PRACTICE QUESTIONS ON MATRICES AND DETERMINANTS

Problem 1 :

If |adj(adj A)| = |A|⁹, then the order of the square matrix A is

(1)  3     (2)  4     (3)  2    (4)  5

Solution :

Given, |adj(adj A)| = |A|⁹

To find :

The order of the square matrix A.

Suppose that order of the square matrix A is n.

|adj A| = |A|^{n - 1}

|adj(adj A)| = |adj A|^{n - 1}

= |A|^{(n - 1)}

|A|^{(n - 1)^2} = |A|⁹

(n - 1)² = 9

Take square root on both sides.

√(n - 1)²  = √9

n - 1  = 3

n = 3 + 1

n = 4

The order of the square matrix A is 4.

So, option 2) is correct.

Problem 2 :

If A is a 3 × 3 non - singular matrix such that AA^T = A^T A and B = A^-1 A^T, then BB^T = 

(1)  A     (2)  B     (3)  I₃    (4)  B^T

Solution :

Given, A is a 3 × 3 non - singular matrix such that AA^T = A^T A and B = A^-1 A^T

To find : BB^T

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

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

=  A^-1 A^T (A^-1)^T A

=   A^-1 A A^T (A^-1)^T 

=   A^-1 A A^T (A^T)^-1

=  A^T (A^-1)^T 

 =  (AA^-1)^T 

= I^T

= I

Hence, I = I₃

So, option 3) is correct.

Problem 3 :

(1)  1/3      (2)  1/9    (3)  1/4      (4)  1

Solution :

Hence, the answer is 1/9.

So, option 2) is correct.

Problem 4 :

Solution :

Problem 5 :

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

Solution :

So, option 4) is correct.

Problem 6 :

1) -40     2) -80     3) -60      4)-20

Solution :

So, option 2) is correct.