Let $A$ and $B$ are $3 \times 3$ 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:

JEE Main · 2021 · 24 Feb 2021 · Shift 2 · Matrices

Let

A

and

B

are

3 \times 3

real matrices such that

A

is symmetric matrix and

B

is skew- symmetric matrix. Then the system of linear equations

(A^{2} B^{2} - B^{2} A^{2}) X = O

, where

X

is a

3 \times 1

column matrix of unknown variables and

O

is a

3 \times 1

null matrix has:

Answer: (C)

Since

B^{T} = - B

(B^{2})^{T} = B^{T} B^{T} = (- B) (- B) = B^{2}

, and

(A^{2})^{T} = A^{2}

because

A

is symmetric. Hence

(A^{2} B^{2})^{T} = (B^{2})^{T} (A^{2})^{T} = B^{2} A^{2}

, so with

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

M^{T} = B^{2} A^{2} - A^{2} B^{2} = - M,

making

M

skew-symmetric of order

3

. Then

det M = det M^{T} = det (- M) = (- 1)^{3} det M

, so

det M = 0

and the homogeneous system

M X = O

admits non-trivial solutions besides

X = O

. The system has infinitely many solutions.

JEE Main · 2021 · 24 Feb 2021 · Shift 2 · Single correct option