# Let A and B be 3x3 real matrices such

Question:

Let $\mathrm{A}$ and $\mathrm{B}$ be $3 \times 3$ real matrices such that $\mathrm{A}$ is symmetric matrix and $\mathrm{B}$ 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 $\mathrm{O}$ is a $3 \times 1$ null matrix, has :

1. (1) a unique solution

2. (2) exactly two solutions

3. (3) infinitely many solutions

4. (4) no solution

Correct Option: , 3

Solution:

$A^{\top}=A, B^{\top}=-B$

Let $A^{2} B^{2}-B^{2} A^{2}=P$

$P^{\top}=\left(A^{2} B^{2}-B^{2} A^{2}\right)^{\top}=\left(A^{2} B^{2}\right)^{\top}-\left(B^{2} A^{2}\right)^{\top}$

$=\left(B^{2}\right)^{\top}\left(A^{2}\right)_{0}^{\top}-\left(A^{2}\right)^{\top}\left(B^{2}\right)^{\top}$

$=B^{2} A^{2}-A^{2} B^{2}$

$\Rightarrow \mathrm{P}$ is skew-symmetric matrix

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

$\therefore \quad a y+b z=0$

$-a x+c z=0$

$-b x-c y=0$

From equation $1,2,3$

$\Delta=0 \& \Delta_{1}=\Delta_{2}=\Delta_{3}=0$

$\therefore$ equation have infinite number of solution