Let A and B be $3 \times 3$ real matrices such that A is symmetric matrix and $B$ is skew-symmetric matrix.…
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 $\left(A^2 B^2-B^2 A^2\right) X=0$. where $X$ is $3 \times 1$ column matrix of unknown variables and O is a $3 \times 1$ null matrix, has
a unique solution
exactly two solutions
no solution
infinitely many solutions
Solution
Let $P=A^2 B^2-B^2 A^2$
$\begin{aligned}
\therefore \quad \mathrm{P}^{\mathrm{T}} & =\left(\mathrm{A}^2 \mathrm{~B}^2-\mathrm{B}^2 \mathrm{~A}^2\right)^{\mathrm{T}} \\
& =\left(\mathrm{A}^2 \mathrm{~B}^2\right)^{\mathrm{T}}-\left(\mathrm{B}^2 \mathrm{~A}^2\right)^{\mathrm{T}} \\
& =\left(\mathrm{B}^2\right)^{\mathrm{T}}\left(\mathrm{~A}^2\right)^{\mathrm{T}}-\left(\mathrm{A}^2\right)^{\mathrm{T}}\left(\mathrm{~B}^2\right)^{\mathrm{T}} \\
& =\mathrm{B}^2 \mathrm{~A}^2-\mathrm{A}^2 \mathrm{~B}^2 \quad \cdots\left[\because \mathrm{~A}^{\mathrm{T}}=\mathrm{A} \text { and } \mathrm{B}^{\mathrm{T}}=-\mathrm{B}\right] \\
& =-\left(\mathrm{A}^2 \mathrm{~B}^2-\mathrm{B}^2 \mathrm{~A}^2\right) \\
& =-\mathrm{P}
\end{aligned}$
$\therefore \quad P$ is a skew - symmetric matrix.
$\therefore \quad \operatorname{det}(\mathrm{P})=0$
$\therefore \quad$ The given system of equations has infinitely many solutions.