Solve this following

Question:

Let $\mathrm{A}$ and $\mathrm{B}$ be $3 \times 3$ real matrices such that $\mathrm{A}$ is symmetric matrix and $B$ is skew-symmetric matrix. Then the system of linear equations $\left(\mathrm{A}^{2} \mathrm{~B}^{2}-\mathrm{B}^{2} \mathrm{~A}^{2}\right) \mathrm{X}=\mathrm{O}$, where $\mathrm{X}$ is a $3 \times 1$ column matrix of unknown variables and $\mathrm{O}$ is a $3 \times 1$ null matrix, has:

 

  1. no solution

  2. exactly two solutions

  3. infinitely many solutions

  4. a unique solution


Correct Option: , 3

Solution:

Let $\mathrm{A}^{\mathrm{T}}=\mathrm{A}$ and $\mathrm{B}^{\mathrm{T}}=-\mathrm{B}$

$\mathrm{C}=\mathrm{A}^{2} \mathrm{~B}^{2}-\mathrm{B}^{2} \mathrm{~A}^{2}$

$\mathrm{C}^{\mathrm{T}}=\left(\mathrm{A}^{2} \mathrm{~B}^{2}\right)^{\mathrm{T}}-\left(\mathrm{B}^{2} \mathrm{~A}^{2}\right)^{\mathrm{T}}$

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

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

$C^{T}=-C$

$\mathrm{C}$ is skew symmetric.

So $\operatorname{det}(C)=0$

so system have infinite solutions.

 

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