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Let $A$ and $B$ be any two $3 \times 3$ symmetric and skew symmetric matrices respectively. Then which of the following is $NOT$ true?
$A ^{4}- B ^{4}$ is a symmetric matrix
$AB - BA$ is a symmetric matrix
$B ^{5}- A ^{5}$ is a skew-symmetric matrix
$AB + BA$ is a skew-symmetric matrix
Solution
Given that $A^{T}=A, B^{T}=-B$
$C =A^{4}-B^{4}$
$C^{ T }=\left( A ^{4}- B ^{4}\right)=\left( A ^{4}\right)^{ T }-\left( B ^{4}\right)^{ T }= A ^{4}- B ^{4}= C$
$C = AB – BA$
$C ^{ T }=( AB – BA )^{ T }=( AB )^{ T }-( BA )^{ T }$
$= B ^{ T } A ^{ T }- A ^{ T } B ^{ T }=- BA + AB = C$
$C = B ^{5}- A ^{5}$
$C ^{ T }=\left( B ^{5}- A ^{5}\right)^{ T }=\left( B ^{ S }\right)^{ T }-\left( A ^{5}\right)^{ T }=- B ^{5}- A ^{5}$
$C = AB + BA$
$C ^{ T }=( AB + BA )^{ T }=( AB )^{ T }+( BA )^{ T }$
$=- BA – AB =- C$
$\therefore \text { Option } C \text { is not true. }$
