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  • Let A and B be real matrices such that A is symmetric matrix and B is skew - symmetric matrix. Then the system of linear equations <img alt="\left ( A^{2}B^{2

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 = O, where X is a 3 \times 1 column matrix of unknown variables and O is a 3 \times 1 null matrix, has :
Option: 1 infinitely many solutions
Option: 2 no solution
Option: 3 a unique solution
Option: 4 exactly two solutions

Answers (1)

best_answer

 

\\ \text {Let } A^{T}=A \text { and } B^{T}=-B \\\\ C=A^{2} B^{2}-B^{2} A^{2} \\ \\C^{T}=\left(A^{2} B^{2}\right)^{T}-\left(B^{2} A^{2}\right)^{T} \\ \\C^{T}=\left(B^{2}\right)^{T}\left(A^{2}\right)^{T}-\left(A^{2}\right)^{T}\left(B^{2}\right)^{T}

\\ \\C^{T}= B^{2} A^{2}-A^{2} B^{2} \\\\ C^{T}=-C

C is skew symmetric.

So \operatorname{det}(C)=0

so system have infinite solutions.

Posted by

Suraj Bhandari

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