1.

If A and B are symmetric matrices of same order, prove that (i) AB + BA is a symmetric matrix. (ii) AB – BA is a skew-symmetric matrix.

Answer»

Given A and B are symmetric matrices 

⇒ – AT = A and BT = B 

(i) To prove AB + BA is a symmetric matrix. 

Proof: Now (AB + BA)T = (AB)T + (BA)T = BT AT + AT BT 

= BA + AB = AB + BA 

i.e. (AB + BA)T = AB + BA

⇒ (AB + BA) is a symmetric matrix. 

(ii) To prove AB – BA is a skew symmetric matrix. 

Proof: (AB – BA)T = (AB)T – (BA)T = BT AT – AT BT = BA – AB 

i.e. (AB – BA)T = – (AB – BA) 

⇒ AB – BA is a skew symmetric matrix.


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