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QUESTION: 1

Inverse of a matrix A exists, if

Solution:

Only non-singular matrices have inverses.

Suppose that A is invertible. This means that we have the inverse matrix A^{−1} of A.

Consider the equation Ax=0. We show that this equation has only zero solution.

Multiplying it by A^{−1} on the left, we obtain

A^{−1} Ax = A^{-1} 0

⇒ x = 0

QUESTION: 2

If A = , and |3A| = k|A|, then the value of k is

Solution:

|kA| = k^{n}|A|

|3A| = (3)^{n} |A|

= 27|A| = k|A|

k = 27

QUESTION: 3

Find the value of

Solution:

**Correct Answer :- d**

**Explanation:- 3(6-6) -2(6-9) +3(4-6)**

**= 3(0) + 6 - 6**

**= 0**

QUESTION: 4

If A =, then adj A is:

Solution:

QUESTION: 5

The cofactor of an element 9 of the determinant is :

Solution:

Co factor of element 9 will be {(6 * 8) -(-5 * 0)}

= 48

QUESTION: 6

The minor M_{ij} of an element a_{ij} of a determinant is defined as the value of the determinant obtained after deleting the

Solution:

A minor, Mij, of the element aij is the determinant of the matrix obtained by deleting the ith row and jth column.

QUESTION: 7

The cofactor of 5 in

Solution:

A21 = {(0, -1) (3,4)}

⇒ (0 - (-3)) = 3

A21 = (-1)^{(2+1)} (3)

= -3

QUESTION: 8

If , then the relation between x and y is

Solution:

½{(0,0,1) (1,3,1) (x,y,1)} = 0

{(0,0,1) (1,3,1) (x,y,1)} = 0/(½)

{(0,0,1) (1,3,1) (x,y,1)} = 0

0(3-y) -0(1-x) +1(y-3x) = 0

=> y - 3x = 0

=> y = 3x

QUESTION: 9

If matrix A = and A^{2} + aA + b = O, then the values of a and b are:

Solution:

QUESTION: 10

If and A_{ij} are cofactors of a_{ij}, then

Solution:

Δ = Sum of products of element of row(or column) with their corresponding co-factors.

Δ = a11 A11 + a21 A21 + a31 A31

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