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Exercise - 3.3

Matrices

**Question-1 :-**
Find the transpose of each of the following matrices:

(i)

(ii)

(iii)

**Question-2 :-**
(i) (A + B)′ = A′ + B′,

(ii) (A – B)′ = A′ – B′

(i)

(ii)

**Question-3 :-**
(i) (A + B)′ = A′ + B′,

(ii) (A – B)′ = A′ – B′

(i)

(ii)

**Question-4 :-**

**Question-5 :-**
For the matrices A and B, verify that (AB)′ = B′A′, where

(i)

(ii)

**Question-6 :-**

(i)

(ii)

**Question-7 :-**

(i)

(ii)

**Question-8 :-**
(i) (A + A′) is a symmetric matrix

(ii) (A – A′) is a skew symmetric matrix

(i)

(ii)

**Question-9 :-**

**Question-10 :-**
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:

(i)

(ii)

(iii)

(iv)

**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

A and B are symmetric matrices, therefore, we have: A' = A, B' = B .........(I) Consider (AB - BA)' = (AB)' - (BA)' [(A - B)' = A' - B'] = B'A' - A'B' [(AB)' = B'A'] = AB - BA [By equation I] Thus, (AB − BA) is a skew-symmetric matrix. The correct answer is A.

**Question-12 :-**
(A) π/6 (B) π/3 (C) π (D) 3π/2

The correct answer is B.

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