Question

If A and B are symmetric matrices of the same order then (AB – BA) is always 

A. a symmetric matrix 

B. a skew-symmetric matrix 

C. a zero matrix 

D. an identity matrix

Answer 1

Given A and B are symmetric matrices 

A’ = A --- 1 

B’ = B ---- 2 

Now (AB – BA)’ = (AB)’ – (BA)’ 

= B’A’ – A’B’ 

[∵(AB) ’ = B’A’ ] 

= BA – AB [Using 1 and 2] 

∴(AB – BA)’ = - (AB - BA) 

AB-BA is a skew symmetric matrix.