Let A and B be 3 × 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix.

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3 and 4 .Determinants and Matrices

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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 ....... .

A

no solution

B

exactly two solutions

C

infinitely many solutions

D

a unique solution

(JEE MAIN-2021)

Solution

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

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

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

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

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

$C ^{ T }=- C$

$C$ is skew symmetric.

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

so system have infinite solutions.

Standard 12

Mathematics

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