Question

Let $A$ and $B$ be $3 \times 3$ real matrices such that $A$ is symmetric matrix and $B$ is skew-symmetric matrix. Then the system of linear equations $\left( A ^{2} B ^{2}- B ^{2} A ^{2}\right) X = O ,$ where $X$ is a $3 \times 1$ column matrix of unknown variables and $O$ is a $3 \times 1$ null matrix, has :

• A. no solution
• B. exactly two solutions
• C. infinitely many solutions
• D. a unique solution

Answer 1

Answer: (C)

Let $A ^{ T }= A \text { and } B ^{ T }=- B$
$C = A ^{2} B ^{2}- B ^{2} A ^{2}$
$C ^{ T }=\left( A ^{2} B ^{2}\right)^{ T }-\left( B ^{2} A ^{2}\right)^{ T }$
$=\left( B ^{2}\right)^{ T }\left( A ^{2}\right)^{ T }-\left( A ^{2}\right)^{ T }\left( B ^{2}\right)^{ T }$
$= B ^{2} A ^{2}- A ^{2} B ^{2}$
$C ^{ T }=- C$
C is skew symmetric.
So det $(C)=0$
So system have infinite solutions.