NCERT Solutions for Class 12 Maths Chapter 3 Matrices: Matrix is a Latin word which means "womb", an environment where something grows. In this article, you will find NCERT solutions for class 12 maths chapter 3 matrices. Matrix is an array of numbers or mathematical objects for which some operations like addition and multiplication are defined. Matrices is an important and powerful tool in mathematics and it's basically introduced to solve simultaneous linear equations. Matrices have a lot of applications like solving the system of linear equations, doing transformations of one vector space to another, etc. It has applications in engineering, for example, the image or video that you see are matrices of color intensities. In this chapter, you will learn about matrices and it's properties. In order to get the solutions of NCERT for class 12 maths chapter 3 matrices, you can go through this article. Important topics that are going to be discussed in this chapter are matrices, the order of a matrix, types of matrices, equality of matrices, operations like addition multiplication on matrices, symmetric and skew-symmetric matrices, etc. In CBSE NCERT solutions for class 12 maths chapter 3 matrices article, questions from all topics are covered. The practice is very important to command over any chapter of CBSE maths, so you should try to solve every problem on your own. If you are not able to solve, you can take the help of NCERT solutions for class 12 maths chapter 3 matrices, which are explained in a detailed manner. Check all NCERT solutions from class 6 to 12 at a single place which will be helpful in order to learn CBSE science and maths.
The topic algebra which contains two topics matrices and determinants which has 13 % weightage in the maths CBSE 12th board final examination, which means you will see 10 marks questions from these two chapters matrices, and determinants in 12th board final exam out of 80 marks. Matrix is a very important chapter from the exam point of view, also from the application point of view, it is very important in further studies like engineering. In this chapter, there are 4 exercises with 62 questions. All these questions are prepared and explained in the NCERT solutions for class 12 maths chapter 3 matrices article. These solutions of NCERT will help you to understand the concept more easily, and perform well in the CBSE 12th board exam.
What are matrices?
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. 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 chapter 3 matrices in a better way total of 28 solved examples are given and also fto practice more, at the end of the chapter, 15 questions are given in the miscellaneous exercise.
Topics and sub-topics of NCERT Grade 12 Maths Chapter 3 Matrices
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
NCERT solutions for class 12 maths chapter 3 matrices- Solved exercise questions
Question 6(ii) Find the values of x, y and z from the following equations:
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
Question 6. Find the values of x, y and z from the following equations
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
(D) None of these
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 correct.
Question 2(iii). Compute the following:
(iii) The addition of given three by three matrix is performed as follows
Question 6. Simplify .
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
(i) The given matrices are
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
Multiply equation 1 by 3 and equation 2 by 2 and subtract them,
Putting value of Y in equation 1 , we get
Question 8. Find X, if and
Substituting the value of Y in the above equation
Question 9. Find x and y, if
Now equating LHS and RHS we can write the following equations
Question 11. If , find the values of x and y.
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 13. If , show that .
To prove :
Hence, we have L.H.S. = R.H.S i.e. .
Question 14(i). Show that
Hence, the right-hand side not equal to the left-hand side, that is
Question 14(ii). Show that
To prove the following multiplication of three by three matrices are not equal
Hence, i.e. .
Question 15. Find, if
First, we will find ou the value of the square of matrix A
Question 16. If prove that .
First, find the square of matrix A and then multiply it with A to get the cube of matrix A
Hence, L.H.S = R.H.S
Question 21 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.
Q21. The restriction on n, k and p so that PY + WY will be defined are:
(B) k is arbitrary,
(C) p is arbitrary,
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.
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.
CBSE NCERT Solutions for class 12 maths chapter -3 Matrices: Exercise 3.3
Question 1(i). Find the transpose of each of the following matrices:
The transpose of the given matrix is
Transpose is obtained by interchanging the rows and columns of matrix
Question 2(i). If and , then verify
Thus we find that the LHS is equal to RHS and hence verified.
Question 2(ii). If and , then verify
Hence, L.H.S = R.H.S. so verified that
To prove :
Hence, L.H.S =R.H.S
so it is verified that .
Question 6(i). If , then verify that
By interchanging rows and columns we get transpose of A
Question 6(ii). If , then verify that
By interchanging columns and rows of the matrix A we get the transpose of A
Question 9. Find and , when
the transpose of the matrix is obtained by interchanging rows and columns
Thus, is a symmetric matrix.
Thus, is a skew-symmetric matrix.
Represent A as the sum of B and C.