PROBLEMS ON FINDING INVERSE AND ADJOINT OF A MATRIX

Problem 1 :

(1) 15     (2) 12     (3) 14     (4) 11

Solution:

| P | = 1(-6 - 0) - x(-2 - 0) + 0 (4 - 6)

| P | = -6 + 2x

| P | = 2x - 6

Now, adj A = P

|adj A| = | P |

| A |² = | P |

| P | = 16

2x - 6 = 16

2x = 22

x = 11

So, option (4) is correct.

Problem 2 :

(1) 0     (2) -2     (3) -3     (4) -1

Solution:

a₂₃ = -2/2 = -1

Problem 3 :

If A, B and C are invertible matrices of some order, then which one of the following is not true?

(1) adj A = | A | A^-1               (2) adj (AB) = (adj A)(adj B)

(3) det A^-1 = (det A)^-1           (4) (ABC)^-1 = C^-1B^-1A^-1

Solution:

adj (AB) = (adj A)(adj B)

So, option (2) is correct.

Problem 4 :

Solution:

So, option (1) is correct.

Problem 5 :

If A^T A^-1 is symmetric, then A² = 

(1) A^-1     (2) (A^T)²     (3) A^T     (4) (A^-1)²

Solution:

(A^TA^-1)^T = A^TA^-1

(A^-1)^T(A^T)^T = A^TA^-1

(A^-1)^T(A) = A^TA^-1

Multiply by A on both sides,

(A^-1)^T(A)(A)= A^TA^-1(A)

(A^-1)^T(A²) = A^TI

Multiply by A^T on both sides,

(A^T)(A^-1)^T(A²) = (A^T)A^TI

(A^T)(A^T)^-1(A²) = (AT)²I

IA² = (A^T)²I

A² = (A^T)²

So, option (2) is correct.

Problem 6 :

Solution:

So, option (4) is correct.