Let A and B be any two 33 symmetric and skew-symmetric matrices respectively.

Question

Let $A$ and $B$ be any two $3	imes3$ symmetric and skew-symmetric matrices respectively. Then which of the following is NOT true?

Accepted Answer

(A) $M = {A^4} - {B^4}$

$${M^T} = {({A^4} - {B^4})^T} = {({A^T})^4} - {({B^T})^4}$$

$$ = {A^4} - {( - B)^4} = {A^4} - {B^4} = M$$

(B) $M = AB - BA$

$${M^T} = {(AB - BA)^T} = {(AB)^T} - {(BA)^T}$$

$= {B^T}{A^T} - {A^T}{B^T}$

$=  - BA - A( - B)$

$= AB - BA = M$

(C) $M = {B^5} - {A^5}$

$${M^T} = {({B^T})^5} - {({A^T})^5} =  - ({B^5} + {A^5}) \ne  - M$$

(D) $M = AB + BA$

${M^T} = {(AB)^T} + {(BA)^T}$

$$ = {B^T}{A^T} + {A^T}{B^T} =  - BA - AB =  - M$$

Solutions · 1