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NCERT Solutions for Class 12 Maths Chapter 3 Matrices are provided here. Matrices is an important and powerful tool in mathematics and it's basically introduced to solve simultaneous linear equations. maths chapter 3 class 12 includes the type of matrices, operations of matrices, and many more concepts. These concepts are used to solve challenging problems of 3d geometry, and determinants Therefore matrices class 12 solutions become important to get command of concepts. The NCERT Solutions are prepared by expert team according the latest syllabus of CBSE 2023. Also, Students can refer matrices class 12 NCERT solutions which covers all the questions of NCERT Books for Class 12 Maths.
The practice from NCERT solutions for class 12 maths Chapter 3 Matrices is very important to score well in academics as well as in competitive exams. Matrices class 12 also includes exercises you can also solve to build hold on the concepts. All the questions in NCERT maths class 12 chapter 3 are explained in a detailed manner. You can also refer to class 12 maths ncert solutions chapter 3 matrices by clicking on the link NCERT solutions for Class 12.
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Matrix Definition and Properties:
A matrix is an ordered rectangular array of numbers or functions.
A matrix of order m × n consists of m rows and n columns.
The order of a matrix is written as m × n, where m is the number of rows and n is the number of columns.
A matrix is called a square matrix when m = n.
A diagonal matrix A = [aij]m×m has aij = 0 when i ≠ j.
A scalar matrix A = [aij]n×n has aij = 0 when i ≠ j, aij = k (where k is a constant)
when i = j.
An identity matrix A = [aij]n×n has aij = 1 when i = j and aij = 0 when i ≠ j.
A zero matrix contains all its elements as zero.
A column matrix is of the form [A]n × 1.
A row matrix is of the form [A]1 × n.
Equality of Matrices:
Two matrices A and B are equal (A = B) if they have the same order and aij = bij for all the corresponding values of i and j.
Operations on Matrices:
Matrix Addition:
If A = [aij]m × n and B = [bij]m × n, then A + B = [aij + bij]m × n.
Matrix Subtraction:
If A = [aij]m × n and B = [bij]m × n, then A - B = [aij - bij]m × n.
As per latest 2024 syllabus. Physics formulas, equations, & laws of class 11 & 12th chapters
Multiplication of a Matrix by Scalar:
Let A = [aij]m × n be a matrix and k is a scalar, then kA is obtained by multiplying each element of A by the scalar k, i.e., kA = [kaij]m × n.
Multiplication of Matrices:
Let A be an m × p matrix, and B be a p × n matrix. Their product AB is defined if the number of columns in A is equal to the number of rows in B. The resulting matrix is an m × n matrix, and the elements are calculated as follows: (AB)ij = Σ(ai * bj), where the sum is taken over all values of p.
Transpose of a Matrix:
The transpose of a matrix A, denoted as AT, is obtained by interchanging its rows and columns.
Symmetric and Skew-Symmetric Matrices:
A matrix A is symmetric if A = AT (i.e., it is equal to its transpose).
A matrix A is skew-symmetric if AT = -A (i.e., the transpose of A is equal to the negative of A).
Elementary Operation or Transformation of a Matrix:
Elementary row operations include:
Interchanging any two rows.
Multiplying a row by a non-zero scalar.
Adding or subtracting a multiple of one row from another row.
Inverse of a Matrix by Elementary Operations:
You can find the inverse of a matrix using elementary row operations. If the matrix A is invertible, you can transform it into the identity matrix I through row operations on an augmented matrix [A | I], where I is the identity matrix of the same order as A. If this process is successful, the resulting matrix on the left will be I, and the matrix on the right will be the inverse of A.
Free download Matrices Class 12 Questions And Answers for CBSE Exam.
Matrices Class 12 Questions And Answers: Exercise 3.1
Question:1(i). In the matrix , write:
Answer:
(i) The order of the matrix = number of row number of columns .
Question 1(iii). In the matrix , write:
Write the elements a 13 , a 21 , a 33 , a 24 , a 23
Answer:
(iii) An element implies the element in raw number i and column number j.
Question 2. If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?
Answer:
A matrix has 24 elements.
The possible orders are :
.
If it has 13 elements, then possible orders are :
.
Question 3. If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?
Answer:
A matrix has 18 elements.
The possible orders are as given below
If it has 5 elements, then possible orders are :
.
Question 4(i). Construct a 2 × 2 matrix, whose elements are given by:
Answer:
(i)
Each element of this matrix is calculated as follows
Matrix A is given by
Question 4(ii). Construct a 2 × 2 matrix, , whose elements are given by:
Answer:
A 2 × 2 matrix,
(ii)
Hence, the matrix is
Question 4(iii). Construct a 2 × 2 matrix, , whose elements are given by:
Answer:
(iii)
Hence, the matrix is given by
Question 5(i). Construct a 3 × 4 matrix, whose elements are given by:
Answer:
(i)
Hence, the required matrix of the given order is
Question 5(ii) Construct a 3 × 4 matrix, whose elements are given by:
Answer:
A 3 × 4 matrix,
(ii)
Hence, the matrix is
Question 6(i). Find the values of x, y and z from the following equations:
Answer:
(i)
If two matrices are equal, then their corresponding elements are also equal.
Question 6(ii) Find the values of x, y and z from the following equations:
Answer:
(ii)
If two matrices are equal, then their corresponding elements are also equal.
Solving equation (i) and (ii) ,
solving this equation we get,
Putting the values of y, we get
And also equating the first element of the second raw
,
Hence,
Question 6(iii) Find the values of x, y and z from the following equations
Answer:
(iii)
If two matrices are equal, then their corresponding elements are also equal
subtracting (2) from (1) we will get y=4
substituting the value of y in equation (3) we will get z=3
now substituting the value of z in equation (2) we will get x=2
therefore,
, and
Question 7. Find the value of a, b, c and d from the equation:
Answer:
If two matrices are equal, then their corresponding elements are also equal
Solving equation 1 and 3 , we get
Putting the value of a in equation 2, we get
Putting the value of c in equation 4 , we get
Question 8. is a square matrix, if
Answer:
A square matrix has the number of rows and columns equal.
Thus, for to be a square matrix m and n should be equal.
Option (c) is .
Question 9. Which of the given values of x and y make the following pair of matrices equal
Answer:
Given,
If two matrices are equal, then their corresponding elements are also equal
Here, the value of x is not unique, so option B is correct.
Question 10. The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:
Answer:
Total number of elements in a 3 × 3 matrix
If each entry is 0 or 1 then for every entry there are 2 permutations.
The total permutations for 9 elements
Thus, option (D) is correct.
Class 12 Maths Chapter 3 Question Answer: Exercise 3.2
Question 1(i) Let , ,
Answer:
(i) A + B
The addition of matrix can be done as follows
Question 1(iii) Let , ,
Answer:
(iii) 3A - C
First multiply each element of A with 3 and then subtract C
Question 2(ii). Compute the following:
Answer:
(ii) The addition operation can be performed as follows
Question 2(iii). Compute the following:
Answer:
(iii) The addition of given three by three matrix is performed as follows
Question 3(i). Compute the indicated products.
Answer:
(i) The multiplication is performed as follows
Question 3(ii). Compute the indicated products.
Answer:
(ii) the multiplication can be performed as follows
Question 3(iii). Compute the indicated products.
Answer:
(iii) The multiplication can be performed as follows
Question 3(iv). Compute the indicated products.
Answer:
(iv) The multiplication is performed as follows
Question 3(vi). Compute the indicated products.
Answer:
(vi) The given product can be computed as follows
Question 4. If , and , then compute (A+B) and (B-C). Also verify that A + (B - C) = (A + B) - C
Answer:
, and
Now, to prove A + (B - C) = (A + B) - C
(Puting value of from above)
Hence, we can see L.H.S = R.H.S =
Question 6. Simplify .
Answer:
The simplification is explained in the following step
the final answer is an identity matrix of order 2
Question 7(i). Find X and Y, if
Answer:
(i) The given matrices are
and
Adding equation 1 and 2, we get
Putting the value of X in equation 1, we get
Question 7(ii). Find X and Y, if
Answer:
(ii) and
Multiply equation 1 by 3 and equation 2 by 2 and subtract them,
Putting value of Y in equation 1 , we get
Question 10. Solve the equation for x, y, z and t, if
Answer:
Multiplying with constant terms and rearranging we can rewrite the matrix as
Dividing by 2 on both sides
Question 11. If , find the values of x and y.
Answer:
Adding both the matrix in LHS and rewriting
Adding equation 1 and 2, we get
Put the value of x in equation 2, we have
Question 12. Given , find the values of x, y, z and w.
Answer:
If two matrices are equal than corresponding elements are also equal.
Thus, we have
Put the value of x
Hence, we have
Question 14(i). Show that
Answer:
To prove:
Hence, the right-hand side not equal to the left-hand side, that is
Question 14(ii). Show that
Answer:
To prove the following multiplication of three by three matrices are not equal
Hence, i.e. .
Question 16. If prove that .
Answer:
First, find the square of matrix A and then multiply it with A to get the cube of matrix A
L.H.S :
Hence, L.H.S = R.H.S
i.e. .
Question 18. If and I is the identity matrix of order 2, show that
Answer:
To prove :
L.H.S :
R.H.S :
Hence, we can see L.H.S = R.H.S
i.e. .
Answer:
Let Rs. x be invested in the first bond.
Money invested in second bond = Rs (3000-x)
The first bond pays 5% interest per year and the second bond pays 7% interest per year.
To obtain an annual total interest of Rs. 1800, we have
(simple interest for 1 year )
Thus, to obtain an annual total interest of Rs. 1800, the trust fund should invest Rs 15000 in the first bond and Rs 15000 in the second bond.
Answer:
Let Rs. x be invested in the first bond.
Money invested in second bond = Rs (3000-x)
The first bond pays 5% interest per year and the second bond pays 7% interest per year.
To obtain an annual total interest of Rs. 1800, we have
(simple interest for 1 year )
Thus, to obtain an annual total interest of Rs. 2000, the trust fund should invest Rs 5000 in the first bond and Rs 25000 in the second bond.
Answer:
The bookshop has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books.
Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively.
The total amount the bookshop will receive from selling all the books:
The total amount the bookshop will receive from selling all the books is 20160.
Q21. The restriction on n, k and p so that PY + WY will be defined are:
(A)
Answer:
P and Y are of order and respectivly.
PY will be defined only if k=3, i.e. order of PY is .
W and Y are of order and respectivly.
WY is defined because the number of columns of W is equal to the number of rows of Y which is 3, i.e. the order of WY is
Matrices PY and WY can only be added if they both have same order i.e = implies p=n.
Thus, are restrictions on n, k and p so that PY + WY will be defined.
Option (A) is correct.
Question 22 Assume X, Y, Z, W and P are matrices of order 2 × n, 3 × k, 2 × p, n × 3 and p × k ,
respectively. Choose the correct answer in Exercises 21 and 22. If n = p , then the order of the matrix is:
(A) p × 2
(B) 2 × n
(C) n × 3
(D) p × n
Answer:
X has of order .
7X also has of order .
Z has of order .
5Z also has of order .
Mtarices 7X and 5Z can only be subtracted if they both have same order i.e = and it is given that p=n.
We can say that both matrices have order of .
Thus, order of is .
Option (B) is correct.
Class 12 Maths Chapter 3 Question Answer: Exercise 3.3
Question 1(i). Find the transpose of each of the following matrices:
Answer:
The transpose of the given matrix is
Question 1(ii). Find the transpose of each of the following matrices:
Answer:
interchanging the rows and columns of the matrix A we get
Question 1(iii) Find the transpose of each of the following matrices:
Answer:
Transpose is obtained by interchanging the rows and columns of matrix
Question 2(i). If and , then verify
Answer:
and
L.H.S :
R.H.S :
Thus we find that the LHS is equal to RHS and hence verified.
Question 2(ii). If and , then verify
Answer:
and
L.H.S :
R.H.S :
Hence, L.H.S = R.H.S. so verified that
.
Question 4. If and , then find
Answer:
:
Transpose is obtained by interchanging rows and columns and the transpose of A+2B is
Question 5(i) For the matrices A and B, verify that , where
Answer:
,
To prove :
Hence, L.H.S =R.H.S
so it is verified that .
Question 5(ii) For the matrices A and B, verify that , where
Answer:
,
To prove :
Heence, L.H.S =R.H.S i.e. .
Question 6(i). If , then verify that
Answer:
By interchanging rows and columns we get transpose of A
To prove:
L.H.S :
Question 6(ii). If , then verify that
Answer:
By interchanging columns and rows of the matrix A we get the transpose of A
To prove:
L.H.S :
Question 7(i). Show that the matrix is a symmetric matrix.
Answer:
the transpose of A is
Since, so given matrix is a symmetric matrix.
Question 7(ii) Show that the matrix is a skew-symmetric matrix.
Answer:
The transpose of A is
Since, so given matrix is a skew-symmetric matrix.
Question 9. Find and , when
Answer:
the transpose of the matrix is obtained by interchanging rows and columns
Question 10(i). Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
Answer:
Let
Thus, is a symmetric matrix.
Let
Thus, is a skew symmetric matrix.
Represent A as sum of B and C.
Question:10(ii). Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
Answer:
Let
Thus, is a symmetric matrix.
Let
Thus, is a skew-symmetric matrix.
Represent A as the sum of B and C.
Question 10(iii). Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
Answer:
Let
Thus, is a symmetric matrix.
Let
Thus, is a skew-symmetric matrix.
Represent A as the sum of B and C.
Question 10(iv). Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
Answer:
Let
Thus, is a symmetric matrix.
Let
Thus, is a skew-symmetric matrix.
Represent A as the sum of B and C.
Question 11 Choose the correct answer in the Exercises 11 and 12.
If A, B are symmetric matrices of same order, then AB – BA is a
(A) Skew symmetric matrix
(B) Symmetric matrix
(C) Zero matrix
(D) Identity matrix
Answer:
If A, B are symmetric matrices then
and
we have,
Hence, we have
Thus,( AB-BA)' is skew symmetric.
Option A is correct.
NCERT solutions for class 12 maths chapter 3 Matrices: Exercise 3.4
Answer:
Use the elementary transformation we can find the inverse as follows
Answer:
Thus we have obtained the inverse of the given matrix through elementary transformation
Answer:
Using elementary transformations
.
Answer:
Answer:
.
Thus the inverse of matrix A is obtained.
Answer:
Use the elementary transformation
.
Answer:
.
Thus the inverse of matrix A is obtained using elementary transformation.
Answer:
Thus using elementary transformation inverse of A is obtained as
.
Answer:
Thus using elementary transformation the inverse of A is obtained as
.
Answer:
.
Thus the inverse of A is obtained using elementary transformation.
Answer:
thus the inverse of matrix A is
.
Answer:
Hence, we can see all the zeros in the second row of the matrix in L.H.S so does not exist.
Answer:
so the inverse of matrix A is
.
Answer:
Hence, we can see all upper values of matirix are zeros in L.H.S so does not exists.
Answer:
Thos the Inverse of A is
. .
Answer:
and
and
and
Thus the inverse of three by three matrix A is
. .
Answer:
and
Thus the inverse of A is obtained as
. .
Question:18 Matrices A and B will be inverse of each other only if
Answer:
We know that if A is a square matrix of order n and there is another matrix B of same order n, such that , then B is inverse of matrix A.
In this case, it is clear that A is inverse of B.
Hence, m atrices A and B will be inverse of each other only if .
Option D is correct .
NCERT solutions for class 12 maths chapter - 3 Matrices: Miscellaneous exercise
Question:1 Let , show that , where I is the identity matrix of order 2 and .
Answer:
Given :
To prove :
For n=1,
The result is true for n=1.
Let result be true for n=k,
Now, we prove that the result is true for n=k+1,
Put the value of in above equation,
Hence, the result is true for n=k+1.
Thus, we have where , .
Question 2. If then show that , .
Answer:
Given :
To prove:
For n=1, we have
Thus, the result is true for n=1.
Now, take n=k,
For, n=k+1,
Thus, the result is true for n=k+1.
Hence, we have , where .
Question 3. If , then prove that , where n is any positive integer.
Answer:
Given :
To prove:
For n=1, we have
Thus, result is true for n=1.
Now, take result is true for n=k,
For, n=k+1,
Thus, the result is true for n=k+1.
Hence, we have , where .
Question 4. If A and B are symmetric matrices, prove that is a skew symmetric matrix.
Answer:
If A, B are symmetric matrices then
and
we have,
Hence, we have
Thus,( AB-BA)' is skew symmetric.
Question 5. Show that the matrix B′AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
Answer:
Let be a A is symmetric matrix , then
Consider,
Replace by
i.e.
Thus, if A is a symmetric matrix than is a symmetric matrix.
Now, let A be a skew-symmetric matrix, then .
Replace by - ,
i.e. .
Thus, if A is a skew-symmetric matrix then is a skew-symmetric matrix.
Hence, the matrix B′AB is symmetric or skew-symmetric according to as A is symmetric or skew-symmetric.
Question 6. Find the values of x , y , z if the matrix satisfy the equation
Answer:
Thus equating the terms elementwise
Question 10(a) A manufacturer produces three products x, y, z which he sells in two markets.
Annual sales are indicated below:
Market Products
I 10,000 2,000 18,000
II 6,000 20,000 8,000
Answer:
The unit sale prices of x, y and z are ` 2.50, ` 1.50 and ` 1.00, respectively.
The total revenue in the market I with the help of matrix algebra can be represented as :
The total revenue in market II with the help of matrix algebra can be represented as :
Hence, total revenue in the market I is 46000 and total revenue in market II is 53000.
Question 10(b). A manufacturer produces three products x, y, z which he sells in two markets.
Annual sales are indicated below:
Market Products
I 10,000 2,000 18,000
II 6,000 20,000 8,000
Answer:
The unit costs of the above three commodities are ` 2.00, ` 1.00 and 50 paise respectively.
The total cost price in market I with the help of matrix algebra can be represented as :
Total revenue in the market I is 46000 , gross profit in the market is
The total cost price in market II with the help of matrix algebra can be represented as :
Total revenue in market II is 53000, gross profit in the market is
Question 11. Find the matrix X so that
Answer:
The matrix given on R.H.S is matrix and on LH.S is matrix.Therefore, X has to be matrix.
Let X be
Taking,
Hence, we have
Matrix X is .
Answer:
A and B are square matrices of the same order such that ,
To prove : ,
For n=1, we have
Thus, the result is true for n=1.
Let the result be true for n=k,then we have
Now, taking n=k+1 , we have
Thus, the result is true for n=k+1.
Hence, we have , .
To prove:
For n=1, we have
Thus, the result is true for n=1.
Let the result be true for n=k,then we have
Now, taking n=k+1 , we have
Thus, the result is true for n=k+1.
Hence, we have and for all .
Question 13 Choose the correct answer in the following questions:
Answer:
Thus we obtained that
Option C is correct.
Question 14. If the matrix A is both symmetric and skew-symmetric, then
(A) A is a diagonal matrix
(B) A is a zero matrix
(C) A is a square matrix
(D) None of these
Answer:
If the matrix A is both symmetric and skew-symmetric, then
and
Hence, A is a zero matrix.
Option B is correct.
Question 15. If A is square matrix such that , then is equal to
Answer:
A is a square matrix such that
(Replace by )
Hence, we have
Option C is correct.
If you are interested in Matrices Class 12 NCERT Solutions exercises then these are listed below.
Matrices Class 12 NCERT Solutions Exercise 3.1
Matrices Class 12 NCERT Solutions Exercise 3.2
Matrices Class 12 NCERT Solutions Exercise 3.3
Matrices Class 12 NCERT Solutions Exercise 3.4
Matrices Class 12 NCERT Solutions Miscellaneous Exercise
Matrix is an array of numbers. Matrix is a mode of representing data to ease calculation and it is one of the most important tools of mathematics because matrices simplify our work to a great extent when compared with other straight forward methods. Exercises are given in chapter 3 maths class 12 students should refer for practice. Matrices class 12 solutions also provided by careers360’s expert team if you are facing problems then you can refer to it as well.
Matrices are used as a representation of the coefficients in the system of linear equations, electronic spreadsheet programs, also used in business and science. For the students to understand NCERT class 12 maths solutions chapter 3 in a better way, a total of 28 solved examples are given to practice more, at the end of the chapter, 15 questions are given in the miscellaneous exercise. For command on concepts you can uses these practice problems of matrices class 12.
3.1 Introduction
3.2 Matrix
3.2.1 Order of a matrix
3.3 Types of Matrices
3.3.1 Equality of matrices
3.4 Operations on Matrices
3.4.1 Addition of matrices
3.4.2 Multiplication of a matrix by a scalar
3.4.3 Properties of matrix addition
3.4.4 Properties of scalar multiplication of a matrix
3.4.5 Multiplication of matrices
3.4.6 Properties of multiplication of matrices
3.5. Transpose of a Matrix
3.5.1 Properties of the transpose of the matrices
3.6 Symmetric and Skew-Symmetric Matrices
3.7 Elementary Operation (Transformation) of a Matrix
3.8 Invertible Matrices
3.8.1 Inverse of a matrix by elementary operations
Topics of NCERT Class 12 Maths Chapter Matrices
The main topics covered in maths chapter 3 class 12 are:
Matrix
It is an ordered representation of numbers and functions, known as elements in a rectangular array. In this class 12 NCERT topics discuss concepts related to the order of a matrix, elements in raw, and columns of a matrix. there are good quality questions in matrix class 12 solutions.
Type of matrices
This ch 1 maths class 12 concerns different types of matrices like column matrix, raw matrix, square matrix, diagonal matrix, scalar matrix, identity matrix, zero matrix, etc. In this ch 3 maths class 12 also discuss comprehensively conditions of equality of matrix. To get command on these concepts you can refer to NCERT solutions for class 12 maths chapter 3.
Operations on matrices
This ch 3 maths class 12 also includes concepts of addition of matrices, multiplication of a matrix by a scalar, negative matrix, difference of matrices. maths class 12 chapter 3 also contains properties of matrix addition that include commutative leas, associative laws, the existence of an additive identity, the existence of additive inverse. Properties of scalar multiplication of matrix elaborated in this chapter. You can refer to class 12 NCERT solutions for questions about these concepts.
multiplication of matrix
Properties of multiplication of matrix that includes associative law, distributive laws, the existence of multiplicative identity. to get command of these concepts you can go through the NCERT solution for class 12 maths chapter 3.
transpose of a matrix
properties of transpose of the matrix are discussed in maths class 12 chapter 3. matrix class 12 solutions include quality questions to understand the concepts.
ch 3 maths class 12 also discuss in detail the concepts of symmetric and skew symmetric matrix, elementary operations (transformation) of a matrix. for questions on these concepts, you can browse NCERT solutions for class 12 chapter 3.
In class 12 NCERT chapter you get an idea of the inverse of a matrix by elementary operation. class 12 NCERT solutions include problems related to these concepts
Topics enumerated in class 12 NCERT are very important and students are suggested to go through all the concepts discussed in the topics. Questions related to all the above topics are covered in the NCERT solutions for class 12 maths chapter 3
Also read,
Chapter 1 | Relations and Functions |
Chapter 2 | Inverse Trigonometric Functions |
Chapter 3 | Matrices |
Chapter 4 | Determinants |
Chapter 5 | Continuity and Differentiability |
Chapter 6 | Application of Derivatives |
Chapter 7 | Integrals |
Chapter 8 | Application of Integrals |
Chapter 9 | Differential Equations |
Chapter 10 | Vector Algebra |
Chapter 11 | Three Dimensional Geometry |
Chapter 12 | Linear Programming |
Chapter 13 | Probability |
NCERT Class 12 Maths solutions chapter 3 covers all the questions given in the textbook. Make sure you have gone through all the important concepts of NCERT class 12th maths Matricesbefore solving the questions.
Firstly try to solve the questions on your own and then check the NCERT class 12 maths chapter 3 solutions.
Along with the NCERT Solutions for Class 12 Maths Chapter 3 Matrices, solve the previous year questions as well.
NCERT Class 12 maths chapter 3 solutions pdf download will be made available soon. Till then you can save this webpage instead of NCERT class 12 maths chapter 3 pdf to practice questions offline.
NCERT solutions are not only important when you are stuck while solving the problems but you will get to know how to answer in the board exam in order to get good marks in the board exam. these class 12 maths ch 3 question answer also make you understand new concepts or in depth understanding of topics and can solve you doubts. in this way ncert solutions becomes important so you can refer ncert textbooks, ncert exercises and ncert solutions.
Matrices, order of a matrix, types of matrices, equality of matrices, operations like addition multiplication on matrices, symmetric and skew-symmetric matrices are the important topics in this chapter. these topics are very important because of use in other supplementary topics like 3d geometry.
Yes, Class 12 Matrices solutions definitely improve the performance on the board exam. It's recommended that students begin by tackling simpler problems before moving on to more challenging ones. Class 12 matrices NCERT solutions will enable them to identify areas where they need improvement. Through repeated practice on their weaker concepts, students can strengthen their skills and perform well on the board exams. Additionally, short-cut tips are provided to assist students in finding easier ways to solve complex problems.
The topic algebra which contains two topics matrices and determinants which has 13 % weightage in the maths CBSE 12th board final examination. students can list out topics according to respective weightage and channelise their energy according to priority of the topics.
Mathematics often has simple chapters that can be enjoyable to study once understood, and matrices are one example. NCERT Class 12 maths matrices focus on the following main topics: matrices, types of matrices, operations on matrices, transpose of a matrix, symmetric and skew symmetric matrices, elementary operations on matrices, and invertible matrices. The material is presented in a straightforward manner to help students achieve good grades in the board exams, regardless of their intelligence. For ease, students can study matrices class 12 ncert pdf both online and offline.
hello,
Yes you can appear for the compartment paper again since CBSE gives three chances to a candidate to clear his/her exams so you still have two more attempts. However, you can appear for your improvement paper for all subjects but you cannot appear for the ones in which you have failed.
I hope this was helpful!
Good Luck
Hello dear,
If you was not able to clear 1st compartment and now you giving second compartment so YES, you can go for your improvement exam next year but if a student receives an improvement, they are given the opportunity to retake the boards as a private candidate the following year, but there are some requirements. First, the student must pass all of their subjects; if they received a compartment in any subject, they must then pass the compartment exam before being eligible for the improvement.
As you can registered yourself as private candidate for giving your improvement exam of 12 standard CBSE(Central Board of Secondary Education).For that you have to wait for a whole year which is bit difficult for you.
Positive side of waiting for whole year is you have a whole year to preparing yourself for your examination. You have no distraction or something which may causes your failure in the exams. In whole year you have to stay focused on your 12 standard examination for doing well in it. By this you get a highest marks as a comparison of others.
Believe in Yourself! You can make anything happen
All the very best.
Hello Student,
I appreciate your Interest in education. See the improvement is not restricted to one subject or multiple subjects and we cannot say if improvement in one subject in one year leads to improvement in more subjects in coming year.
You just need to have a revision of all subjects what you have completed in the school. have a revision and practice of subjects and concepts helps you better.
All the best.
If you'll do hard work then by hard work of 6 months you can achieve your goal but you have to start studying for it dont waste your time its a very important year so please dont waste it otherwise you'll regret.
Yes, you can take admission in class 12th privately there are many colleges in which you can give 12th privately.
The field of biomedical engineering opens up a universe of expert chances. An Individual in the biomedical engineering career path work in the field of engineering as well as medicine, in order to find out solutions to common problems of the two fields. The biomedical engineering job opportunities are to collaborate with doctors and researchers to develop medical systems, equipment, or devices that can solve clinical problems. Here we will be discussing jobs after biomedical engineering, how to get a job in biomedical engineering, biomedical engineering scope, and salary.
Database professionals use software to store and organise data such as financial information, and customer shipping records. Individuals who opt for a career as data administrators ensure that data is available for users and secured from unauthorised sales. DB administrators may work in various types of industries. It may involve computer systems design, service firms, insurance companies, banks and hospitals.
A career as ethical hacker involves various challenges and provides lucrative opportunities in the digital era where every giant business and startup owns its cyberspace on the world wide web. Individuals in the ethical hacker career path try to find the vulnerabilities in the cyber system to get its authority. If he or she succeeds in it then he or she gets its illegal authority. Individuals in the ethical hacker career path then steal information or delete the file that could affect the business, functioning, or services of the organization.
The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.
Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.
Individuals who opt for a career as geothermal engineers are the professionals involved in the processing of geothermal energy. The responsibilities of geothermal engineers may vary depending on the workplace location. Those who work in fields design facilities to process and distribute geothermal energy. They oversee the functioning of machinery used in the field.
Individuals who opt for a career as a remote sensing technician possess unique personalities. Remote sensing analysts seem to be rational human beings, they are strong, independent, persistent, sincere, realistic and resourceful. Some of them are analytical as well, which means they are intelligent, introspective and inquisitive.
Remote sensing scientists use remote sensing technology to support scientists in fields such as community planning, flight planning or the management of natural resources. Analysing data collected from aircraft, satellites or ground-based platforms using statistical analysis software, image analysis software or Geographic Information Systems (GIS) is a significant part of their work. Do you want to learn how to become remote sensing technician? There's no need to be concerned; we've devised a simple remote sensing technician career path for you. Scroll through the pages and read.
The role of geotechnical engineer starts with reviewing the projects needed to define the required material properties. The work responsibilities are followed by a site investigation of rock, soil, fault distribution and bedrock properties on and below an area of interest. The investigation is aimed to improve the ground engineering design and determine their engineering properties that include how they will interact with, on or in a proposed construction.
The role of geotechnical engineer in mining includes designing and determining the type of foundations, earthworks, and or pavement subgrades required for the intended man-made structures to be made. Geotechnical engineering jobs are involved in earthen and concrete dam construction projects, working under a range of normal and extreme loading conditions.
How fascinating it is to represent the whole world on just a piece of paper or a sphere. With the help of maps, we are able to represent the real world on a much smaller scale. Individuals who opt for a career as a cartographer are those who make maps. But, cartography is not just limited to maps, it is about a mixture of art, science, and technology. As a cartographer, not only you will create maps but use various geodetic surveys and remote sensing systems to measure, analyse, and create different maps for political, cultural or educational purposes.
Budget analysis, in a nutshell, entails thoroughly analyzing the details of a financial budget. The budget analysis aims to better understand and manage revenue. Budget analysts assist in the achievement of financial targets, the preservation of profitability, and the pursuit of long-term growth for a business. Budget analysts generally have a bachelor's degree in accounting, finance, economics, or a closely related field. Knowledge of Financial Management is of prime importance in this career.
The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.
Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
An Investment Banking career involves the invention and generation of capital for other organizations, governments, and other entities. Individuals who opt for a career as Investment Bankers are the head of a team dedicated to raising capital by issuing bonds. Investment bankers are termed as the experts who have their fingers on the pulse of the current financial and investing climate. Students can pursue various Investment Banker courses, such as Banking and Insurance, and Economics to opt for an Investment Banking career path.
An underwriter is a person who assesses and evaluates the risk of insurance in his or her field like mortgage, loan, health policy, investment, and so on and so forth. The underwriter career path does involve risks as analysing the risks means finding out if there is a way for the insurance underwriter jobs to recover the money from its clients. If the risk turns out to be too much for the company then in the future it is an underwriter who will be held accountable for it. Therefore, one must carry out his or her job with a lot of attention and diligence.
Individuals in the operations manager jobs are responsible for ensuring the efficiency of each department to acquire its optimal goal. They plan the use of resources and distribution of materials. The operations manager's job description includes managing budgets, negotiating contracts, and performing administrative tasks.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A career as Transportation Planner requires technical application of science and technology in engineering, particularly the concepts, equipment and technologies involved in the production of products and services. In fields like land use, infrastructure review, ecological standards and street design, he or she considers issues of health, environment and performance. A Transportation Planner assigns resources for implementing and designing programmes. He or she is responsible for assessing needs, preparing plans and forecasts and compliance with regulations.
An expert in plumbing is aware of building regulations and safety standards and works to make sure these standards are upheld. Testing pipes for leakage using air pressure and other gauges, and also the ability to construct new pipe systems by cutting, fitting, measuring and threading pipes are some of the other more involved aspects of plumbing. Individuals in the plumber career path are self-employed or work for a small business employing less than ten people, though some might find working for larger entities or the government more desirable.
Individuals who opt for a career as construction managers have a senior-level management role offered in construction firms. Responsibilities in the construction management career path are assigning tasks to workers, inspecting their work, and coordinating with other professionals including architects, subcontractors, and building services engineers.
Urban Planning careers revolve around the idea of developing a plan to use the land optimally, without affecting the environment. Urban planning jobs are offered to those candidates who are skilled in making the right use of land to distribute the growing population, to create various communities.
Urban planning careers come with the opportunity to make changes to the existing cities and towns. They identify various community needs and make short and long-term plans accordingly.
Highway Engineer Job Description: A Highway Engineer is a civil engineer who specialises in planning and building thousands of miles of roads that support connectivity and allow transportation across the country. He or she ensures that traffic management schemes are effectively planned concerning economic sustainability and successful implementation.
Individuals who opt for a career as an environmental engineer are construction professionals who utilise the skills and knowledge of biology, soil science, chemistry and the concept of engineering to design and develop projects that serve as solutions to various environmental problems.
A Naval Architect is a professional who designs, produces and repairs safe and sea-worthy surfaces or underwater structures. A Naval Architect stays involved in creating and designing ships, ferries, submarines and yachts with implementation of various principles such as gravity, ideal hull form, buoyancy and stability.
Orthotists and Prosthetists are professionals who provide aid to patients with disabilities. They fix them to artificial limbs (prosthetics) and help them to regain stability. There are times when people lose their limbs in an accident. In some other occasions, they are born without a limb or orthopaedic impairment. Orthotists and prosthetists play a crucial role in their lives with fixing them to assistive devices and provide mobility.
A career in pathology in India is filled with several responsibilities as it is a medical branch and affects human lives. The demand for pathologists has been increasing over the past few years as people are getting more aware of different diseases. Not only that, but an increase in population and lifestyle changes have also contributed to the increase in a pathologist’s demand. The pathology careers provide an extremely huge number of opportunities and if you want to be a part of the medical field you can consider being a pathologist. If you want to know more about a career in pathology in India then continue reading this article.
Gynaecology can be defined as the study of the female body. The job outlook for gynaecology is excellent since there is evergreen demand for one because of their responsibility of dealing with not only women’s health but also fertility and pregnancy issues. Although most women prefer to have a women obstetrician gynaecologist as their doctor, men also explore a career as a gynaecologist and there are ample amounts of male doctors in the field who are gynaecologists and aid women during delivery and childbirth.
An oncologist is a specialised doctor responsible for providing medical care to patients diagnosed with cancer. He or she uses several therapies to control the cancer and its effect on the human body such as chemotherapy, immunotherapy, radiation therapy and biopsy. An oncologist designs a treatment plan based on a pathology report after diagnosing the type of cancer and where it is spreading inside the body.
The audiologist career involves audiology professionals who are responsible to treat hearing loss and proactively preventing the relevant damage. Individuals who opt for a career as an audiologist use various testing strategies with the aim to determine if someone has a normal sensitivity to sounds or not. After the identification of hearing loss, a hearing doctor is required to determine which sections of the hearing are affected, to what extent they are affected, and where the wound causing the hearing loss is found. As soon as the hearing loss is identified, the patients are provided with recommendations for interventions and rehabilitation such as hearing aids, cochlear implants, and appropriate medical referrals. While audiology is a branch of science that studies and researches hearing, balance, and related disorders.
The hospital Administrator is in charge of organising and supervising the daily operations of medical services and facilities. This organising includes managing of organisation’s staff and its members in service, budgets, service reports, departmental reporting and taking reminders of patient care and services.
For an individual who opts for a career as an actor, the primary responsibility is to completely speak to the character he or she is playing and to persuade the crowd that the character is genuine by connecting with them and bringing them into the story. This applies to significant roles and littler parts, as all roles join to make an effective creation. Here in this article, we will discuss how to become an actor in India, actor exams, actor salary in India, and actor jobs.
Individuals who opt for a career as acrobats create and direct original routines for themselves, in addition to developing interpretations of existing routines. The work of circus acrobats can be seen in a variety of performance settings, including circus, reality shows, sports events like the Olympics, movies and commercials. Individuals who opt for a career as acrobats must be prepared to face rejections and intermittent periods of work. The creativity of acrobats may extend to other aspects of the performance. For example, acrobats in the circus may work with gym trainers, celebrities or collaborate with other professionals to enhance such performance elements as costume and or maybe at the teaching end of the career.
Career as a video game designer is filled with excitement as well as responsibilities. A video game designer is someone who is involved in the process of creating a game from day one. He or she is responsible for fulfilling duties like designing the character of the game, the several levels involved, plot, art and similar other elements. Individuals who opt for a career as a video game designer may also write the codes for the game using different programming languages.
Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.
Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.
A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.
The word “choreography" actually comes from Greek words that mean “dance writing." Individuals who opt for a career as a choreographer create and direct original dances, in addition to developing interpretations of existing dances. A Choreographer dances and utilises his or her creativity in other aspects of dance performance. For example, he or she may work with the music director to select music or collaborate with other famous choreographers to enhance such performance elements as lighting, costume and set design.
A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.
A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.
In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.
Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.
For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.
In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion.
Ever since internet costs got reduced the viewership for these types of content has increased on a large scale. Therefore, a career as a vlogger has a lot to offer. If you want to know more about the Vlogger eligibility, roles and responsibilities then continue reading the article.
Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.
Linguistic meaning is related to language or Linguistics which is the study of languages. A career as a linguistic meaning, a profession that is based on the scientific study of language, and it's a very broad field with many specialities. Famous linguists work in academia, researching and teaching different areas of language, such as phonetics (sounds), syntax (word order) and semantics (meaning).
Other researchers focus on specialities like computational linguistics, which seeks to better match human and computer language capacities, or applied linguistics, which is concerned with improving language education. Still, others work as language experts for the government, advertising companies, dictionary publishers and various other private enterprises. Some might work from home as freelance linguists. Philologist, phonologist, and dialectician are some of Linguist synonym. Linguists can study French, German, Italian.
The career of a travel journalist is full of passion, excitement and responsibility. Journalism as a career could be challenging at times, but if you're someone who has been genuinely enthusiastic about all this, then it is the best decision for you. Travel journalism jobs are all about insightful, artfully written, informative narratives designed to cover the travel industry. Travel Journalist is someone who explores, gathers and presents information as a news article.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.
A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.
A metallurgical engineer is a professional who studies and produces materials that bring power to our world. He or she extracts metals from ores and rocks and transforms them into alloys, high-purity metals and other materials used in developing infrastructure, transportation and healthcare equipment.
An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems.
An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party.
Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack
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