Question 1:

Find the transpose of each of the following matrices:

Answer:

Question 2:

1. (A + B)’ = A’ + B’
2. (A - B)’ = A’ - B’

Answer:

We have

 

1. A + B

   

Hence, we have verified that (A+B)’ = A’ + B’

(ii) A – B

Hence, we have verified that (A - B)’ =A’ – B’.

Question 3:

1. (A + B)’ = A’ + B’
2. (A - B)’ = A’ - B’

Answer:

(i) It is known that A = (A’)’

Therefore, we have:

Thus, we have verified that (A+B)’ = A’ + B’

 (ii)

Thus we have verified that (A-B)’= A’ – B’

 

Question 4:

Answer:

We known that A = (A’)’

 

Question 5:

For the matrices A and B, verify that (AB)′ = B’A’ where

Answer:

Hence, we have verified that (AB)’ = B’A’.

Hence, we have verified that (AB)’=B’A’.

 

Question 6:

Answer:

(i)

 

 

Hence, we have verified that A’A = I

(ii)

Hence, we have verified that A’A=I.

 

Question 7:

(i) Show that the  matrix is a symmetric matrix

(ii) Show that the  matrix is a skew symmetric matrix

Answer

Hence, A is a symmetric matrix.

(ii) We have:

Hence, A is a skew-symmetric matrix.

 

Question 8:

For the matrix  verify that

1. (A+A’) is a symmetric matrix
2. (A-A’) is a skew symmetric matrix

Answer:

 

Hence, (A + A’) is a symmetric matrix.

Hence, (A – A’) is skew-symmetric matrix.

 

Question 9:

Find ½ (A + A’) and ½ (A - A’), When

Answer:

The given matrix is

 

Question 10:

Express the following matrices as the sum of a symmetric and a skew symmetric matrix:

Answer:

(i)

Thus, P = ½ (A+A’) is a symmetric matrix.

Thus, Q = ½ (A-A’) is a skew-symmetric matrix.

Representing A as the sum of P and Q:

(ii)

Thus, P = ½ (A+A’)

Thus, Q = ½ (A-A’) is a skew-symmetric matrix.

Representing A as the sum of P and Q:

(iii)

Thus, P = ½ (A+A’) is a symmetric matrix.

Thus, Q = ½ (A-A’) is a skew-symmetric matrix.

(iv)

Thus, P = ½ (A+A’) is a symmetric matrix.

Thus, Q = ½ (A-A’) is a skew-symmetric matrix.

Representing A as the sum of P and Q:

 

Question 11: If A, B are symmetric matrices of same order, then AB − BA is a

A. Skew symmetric matrix

B. Symmetric matrix

C. Zero matrix

D. Identity matrix

Answer

The correct answer is A.

A and B are symmetric matrices, therefore, we have:

A’ = A and B’=B                              ….(1)

Consider (AB-BA)’ = (AB)’ – (BA)’                           [(A-B)’ = A’ – B’]

= B’A’ – A’B’                     [(AB)’ = B’A’]

= BA – AB                          [by (1)]

= (AB - BA)

∴ (AB - BA)’ = - (AB - BA)

Thus, (AB − BA) is a skew-symmetric matrix.

 

Question 12:

If , then A + A’ = I, if the value of a is

A.  π/6

B. π/3

C. π

D. 3π/2

Answer:

The correct answer is B.

Now, A+A; = I

Comparing the corresponding elements of the two matrices, we have:

2 cos α = 1

⇒ cos α = 1/2 = cos (π/3)

∴ α = (π/3)