Let A and B be 3times3 real matrices where A is symmetric and B is skew-symmetric. The system (A^2B^2 - B^2A^2)X = O (where X is 3times1 and O is the 3times1 zero matrix) has:

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EEJ MAIN Mathematics QUESTION #6946

Question 1

Let $A$ and $B$ be $3\times3$ real matrices where $A$ is symmetric and $B$ is skew-symmetric. The system $(A^2B^2 - B^2A^2)X = O$ (where $X$ is $3\times1$ and $O$ is the $3\times1$ zero matrix) has:

Exactly two solutions
Infinitely many solutions✔️
No solution
A unique solution

Correct Answer Explanation

Let $C = A^2B^2 - B^2A^2$. Since $A$ is symmetric: $A^T=A$, so $(A^2)^T=A^2$. Since $B$ is skew-symmetric: $B^T=-B$, so $(B^2)^T=B^2$.

$C^T = (A^2B^2)^T - (B^2A^2)^T = B^2A^2 - A^2B^2 = -C$

So $C$ is skew-symmetric. A real skew-symmetric $3\times3$ matrix always has determinant 0 (since $\det(C)=\det(C^T)=\det(-C)=-\det(C)$, giving $\det(C)=0$).

Since $\det(C)=0$, the homogeneous system $CX=O$ has infinitely many solutions.

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