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.