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 $(A^2{B}^2-B^2{A}^2)X=O,$ where $X$ is a $3\times 1$ column matrix of unknown variables and $O$ is a $3\times 1$ null matrix, has

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

Answer 1

The correct option is C infinitely many solutions

Given $A^T=A$ and $B^T=-B$
Let $A^2{B}^2-B^2{A}^2=P$
$P^T=(A^2{B}^2-B^2{A}^2{)}^T$
$=(A^2{B}^2{)}^T-(B^2{A}^2{)}^T$
$=\left(B^2{)}^T\right(A^2{)}^T-\left(A^2{)}^T\right(B^2{)}^T$
$=B^2{A}^2-A^2{B}^2$
$\Rightarrow P$ is a skew-symmetric matrix.

$\left[\begin{array}{ccc}0& a& b\\ -a& 0& c\\ -b& -c& 0\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

$\therefore ay+bz=0\cdots \left(1\right)$
$-ax+cz=0\cdots \left(2\right)$
$-bx-cy=0\cdots \left(3\right)$
From equations $\left(1\right),\left(2\right),\left(3\right)$
$\Delta =0$ and ${\Delta }_1={\Delta }_2={\Delta }_3=0$
$\therefore$ Given system of equations has infinite number of solutions.