Matrices(Ex. 3.3)

Exercise 3.3

1. Find the transpose of each of the following matrices:

Taking L.H.S:-(A+B)’

Order of A = Order of B

So the addition of matrix A and B

Taking R.H.S:- A’+B’

L.H.S = R.H.S (Hence Proved……)

(ii) (A-B)’ = A’-B’

Solun:- Taking L.H.S:- (A-B)’

Taking R.H.S:- A’-B’

From Eq.1 and 2:-

L.H.S = R.H.S (Hence Proved…....)

Taking L.H.S:- (A+B)’

From Eq. 1 and 2:- 

Taking R.H.S:- A’+B’

From Eq. 1 and 2:-

L.H.S = R.H.S (Hence Proved……)

(ii) (A-B)’ = A’ - B’

 

So, L.H.S=R.H.S (Hence Proved…....)

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

So (AB)’ = B’A’ (Hence Proved……)

 

So (AB)’ = B’A’ (Hence Proved……)

Solun:- For symmetric matrix A=A’

 So, A’=A 

Thus, A is a symmetric matrix.   

Solun:- For skew-symmetric matrix A = - A’

 So, A’ = - A 

Thus, A is a skew-symmetric matrix.

(i) (A+A’) is a symmetric matrix

So (A+A’) = transpose of (A+A’) = (A+A’)’

So, (A+A’) is a symmetric matrix.

(ii) (A-A)’ is a skew-symmetric matrix

So (A-A’) = - transpose of (A-A’) = - (A-A’)’

So, (A-A’) is a skew-symmetric matrix.

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

So, P=P’ then P is a symmetric matrix.

So Q = - Q’ then Q is a skew symmetric matrix.

Representation of A in P and Q form

⇒ A=P+Q

So, P=P’ then P is a symmetric matrix.

So Q = - Q’ then Q is a skew symmetric matrix.

Representation of A in P and Q form

⇒ A=P+Q

So Q = - Q’ then Q is a skew symmetric matrix.

Representation of A in P and Q form

⇒ A=P+Q

So, P=P’ then P is a symmetric matrix.

So Q = - Q’ then Q is a skew symmetric matrix.

Representation of A in P and Q form

⇒ A=P+Q

Choose the correct answer in the Exercise 11 and 12.

11. If A and 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

Solun:- Given A and B are symmetric matrix

Then A=A’

⇒ B=B’

⇒ (AB-BA)’ = (AB)’ - (BA)’  {We know (A-B)’=A’ - B’ }

⇒ (AB-BA)’ = B’A’ - A’B’      {We know (AB)’ = B’A’}

⇒ (AB-BA)’ = BA - AB   (Given)

⇒ (AB-BA)’ = - (AB-BA)

So, (AB - BA) is the skew-symmetric matrix.

Answer is:- A

(A)π/6  (B) π/3  (C) π (D) 3π/2

(Answer is ……B)

Download pdf of Exercise 3.3

See also:- 

  Exercise 3.2

  Exercise 3.1

  Chapter 3 Notes 

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