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The following system of linear equations   has  Option: 1 infinitely many solutions,  satisfying  Option: 2 infinitely many solutions,  satisfying  Option: 3 no solution Option: 4 only the trivial solution.

System of Homogeneous linear equations -

If ? ≠ 0, then x= 0, y = 0, z = 0 is the only solution of the above system. This solution is also known as a trivial solution.

If ? = 0, at least one of x, y and z are non-zero. This solution is called a non-trivial solution.

Explanation: using equation (ii) and (iii), we have

This is the condition for a system have Non-trivial solution.

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so infinite non-trivial solution exist

now equation (1) + 3 equation (3)

10x - 20z = 0

x = 2z

Correct Option 2

Let   If  then :    Option: 1 Option: 2 Option: 3 Option: 4

Elementary row operations -

Elementary row operations

Row transformation: Following three types of operation (Transformation) on the rows of a given matrix are known as elementary row operation (transformation).

i) Interchange of ith row with jth row, this operation is denoted by

ii) The multiplication of ith row by a constant k (k≠0) is denoted by

iii) The addition of ith row to the elements of jth row multiplied by constant k (k≠0) is denoted by

In the same way, three-column operations can also be defined too.

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Correct Option (3)

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If for some  in R, the intersection of the following three planes  is aline in , then  is equal to :  Option: 1 Option: 2 Option: 3  Option: 4

Cramer’s law -

Cramer’s law for the system of equations in two variables :

We can observe that first row in the numerator of x is of constants and 2nd row in the numerator is of constants, and the denominator is of the coefficient of variables.

We can follow this analogy for the system of equations of 3 variable where third in the numerator of the value of z will be of constant and denominator will be formed by the value of coefficients of the variables.

i) If ? ≠ 0, then the system of equations has a unique finite solution and so equations are consistent, and solutions are

ii) If ? = 0, and any of

Then the system of equations is inconsistent and hence no solution exists.

iii) If all  then

System of equations is consistent and dependent and it has an infinite number of solution.

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The number of all $3\times 3$ matrices A, with enteries from the set $\left \{ -1,0,1 \right \}$ such that the sum of the diagonal elements of $AA^{T}$ is 3, is Option: 1 672 Option: 2 512 Option: 31024 Option: 4 256

Let matrix A be

$\\A=\begin{bmatrix} a &b &c \\ d& e &f \\ g &h & i \end{bmatrix}\\\\\\ A^T=\begin{bmatrix} a &d &g \\ b& e &h \\ c &f & i \end{bmatrix} \\\\\\ \text{trace}(AA^T)=a^2+b^2+c^2+d^2+e^2+f^2+g^2+h^2+i^2=3$

So out of 9 elements, 3 elements must be equal to 1 or −1, and the rest elements must be 0.

Possible cases

$\begin{array}{cc} 0,0,0,0,0,0,1,1,1 & \rightarrow \text{Total possibilities}=^9C_6 \\ 0,0,0,0,0,0,-1,-1,-1 & \rightarrow \text{Total possibilities}=^9C_6 \\ 0,0,0,0,0,0,1,1,-1 & \rightarrow \text{Total possibilities}=^9C_6\times 3 \\ 0,0,0,0,0,0,-1,1,-1 & \rightarrow \text{Total possibilities}=^9C_6\times3 \end{array}$

$\\\\\text{Total number of cases}=^{9}{C}_{6}\times 8=672$

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For which of the following ordered pairs , the system of linear equations is inconsistent ? Option: 1 Option: 2 Option: 3 Option: 4

Solution of System of Linear Equations Using Matrix Method -

let us consider n linear equations in n unknowns, given as below

The above system of equations can be written in matrix form as

Premultiplying equation AX=B by A-1, we get

A-1(AX) = A-1B ⇒ (A-1A)X = A-1B

⇒ IX = A-1B

⇒  X = A-1

⇒

Types of equation :

1. System of equations is non-homogenous:

1. If |A| ≠ 0, then the system of equations is consistent and has a unique solution X = A-1B

2. If |A| = 0 and   (adj A)·B ≠ 0, then the system of equations is inconsistent and has no solution.

3. If |A|  = 0 and   (adj A)·B = 0, then the system of equations is consistent and has infinite number of solutions.

2. System of equations is homogenous:

1. If |A| ≠ 0, then the system of equations has only trivial solution and it has one solution.

2. If |A| = 0 then the system of equations has non-trivial solution and it has an infinite number of solution.

3. If number of equation < number of unknown then it has non-trivial solution.

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Hence, it has infinitely many solutions

For non infinite solution

Correct Option (4)

If the system of linear equations  2x+2ay+az=0 2x+3by+bz=0, 2x+4cy+cz=0, where  are non-zero and distinct ; has a non-zero solution, then : Option: 1 Option: 2  are in AP Option: 3  are in A.P. Option: 4 are in G.P.

Cramer’s law -

Cramer’s law for the system of equations in two variables :

We can observe that first row in the numerator of x is of constants and 2nd row in the numerator is of constants, and the denominator is of the coefficient of variables.

We can follow this analogy for the system of equations of 3 variable where third in the numerator of the value of z will be of constant and denominator will be formed by the value of coefficients of the variables.

i) If ? ≠ 0, then the system of equations has a unique finite solution and so equations are consistent, and solutions are

ii) If ? = 0, and any of

Then the system of equations is inconsistent and hence no solution exists.

iii) If all  then

System of equations is consistent and dependent and it has an infinite number of solution.

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For non zero solutions D = 0

on solving

Correct Option (3)

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The system of linear equations  has : Option: 1 no solution when   Option: 2 infinitely many solutions when  Option: 3 no solution when  Option: 4 a unique solution when

Cramer’s law -

Cramer’s law for the system of equations in two variables :

We can observe that first row in the numerator of x is of constants and 2nd row in the numerator is of constants, and the denominator is of the coefficient of variables.

We can follow this analogy for the system of equations of 3 variable where third in the numerator of the value of z will be of constant and denominator will be formed by the value of coefficients of the variables.

i) If ? ≠ 0, then the system of equations has a unique finite solution and so equations are consistent, and solutions are

ii) If ? = 0, and any of

Then the system of equations is inconsistent and hence no solution exists.

iii) If all  then

System of equations is consistent and dependent and it has an infinite number of solution.

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D = ( + 8) ( 2 – ) for  = 2

= 5[18 – 10] – 2 [48 – 50] + 2 (16 – 30]

= 40 + 4 – 28  0 No solutions for  = 2

Correct Option (1)

If  and  , then  is equal to : Option: 1 Option: 5 Option: 9 Option: 13

Inverse of a Matrix -

A non-singular square matrix “A” is said to be invertible if there exists a non-singular square matrix B such that AB = I = BA, (all matrix are of the same order, they must be for this), then B is called inverse of matrix A.

Inverse of 2 x 2 matrix

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Multiplication of two matrices -

Matrix multiplication:

Two matrices  A and B are conformable for the product AB if the number of columns in A and the number of rows in B is equal. Otherwise, these two matrices will be non-conformable for matrix multiplication. So on that basis,

i) AB is defined only if col(A) = row(B)

ii) BA is defined only if col(B) = row(A)

If

For examples

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On comparing we get

Correct Option (2)

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If the system of linear equations, x+y+z=6 x+2y+3z=10 has more than two solutions, then  is equal to  _______. Option: 1 13 Option: 2 17 Option: 3 21 Option: 4 25

Cramer’s law -

Cramer’s law for the system of equations in two variables :

We can observe that first row in the numerator of x is of constants and 2nd row in the numerator is of constants, and the denominator is of the coefficient of variables.

We can follow this analogy for the system of equations of 3 variable where third in the numerator of the value of z will be of constant and denominator will be formed by the value of coefficients of the variables.

i) If ? ≠ 0, then the system of equations has a unique finite solution and so equations are consistent, and solutions are

ii) If ? = 0, and any of

Then the system of equations is inconsistent and hence no solution exists.

iii) If all  then

System of equations is consistent and dependent and it has an infinite number of solution.

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x + y + z = 6 …….. (1)

x + 2y + 3z = 10 …….. (2)

3x + 2y + z = …….. (3)

from (1) and (2)

if z = 0  x + y = 6 and  x + 2y = 10

y = 4, x = 2

(2, 4, 0)

if y = 0  x + z = 6 and  x + 3z = 10

z = 2 and x = 4

(4, 0, 2)

so, 3x + 2y + z =  must pass through  (2, 4, 0) and  (4, 0, 2)

= 14

and 12 + 2 =

= 1

= 13

Let  and  be two  real matrices such that  where, i, j=1,2,3. if the determinant of B is 81, then the determinant of A is : Option: 1 Option: 2  Option: 3   Option: 4

Matrices, order of matrices, row and column matrix -

A set of numbers (real or complex) or objects or symbols arranged in form of a rectangular array having m rows and n columns and bounded by brackets [?] is called matrix of order m × n, read as m by n matrix.

E.g for m = 2 and n =3, we have   order of this matrix is 2×3

The m by n matrix is represented as :

This representation can be represented in a more compact form as

Where  represents element of ith row and jth column and i = 1,2,...,m; j = 1,2,...,n.

For example, to locate the entry in matrix A identified as aij, we look for the entry in row i, column j. In matrix A, shown below, the entry in row 2, column 3 is a23.

Matrix is only a representation of the symbol, number or object. It does not have any value. Usually, a matrix denoted by capital letters.

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Properties of Determinants - Part 2 -

Property 5

If each element of a row (or a column) of a determinant is multiplied by a constant k, then the value of the determinant is multiplied by k.

For example

Note:

1. By this property, we can take out any common factor from any one row or any one column of a given determinant.

2. If corresponding elements of any two rows (or columns) of a determinant are proportional (in the same ratio), then the determinant value is zero.

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Taking Common  from  and  from

Taking Common  from  and  from

Correct Option (A)