First slide

System of linear equations

Question

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 $(A^{2} B^{2} - B^{2} A^{2}) X = 0$ , where X is a $3 \times 1$ column matrix of unknown variables and O is a $3 \times 1$ null matrix, has

Moderate

A

a unique solution
B

infinitely many solutions
C

no solution
D

exactly two solutions

Solution

$G i v e n A^{T} = A, B^{T} = - B$ given system is homogenous system of equations
Determinant of coefficient matrix is $|A^{2} B^{2} - B^{2} A^{2}|$

$\begin{array}{rcl} C o n s i d e r {(A^{2} B^{2} - B^{2} A^{2})}^{T} & = & {(A^{2} B^{2})^{T} - (B^{2} A^{2})}^{T} \\ = & (B^{2} A^{2}) - (A^{2} B^{2}) \\ = & - (A^{2} B^{2} - B^{2} A^{2}) \end{array}$
Therefore, $(A^{2} B^{2} - B^{2} A^{2})$ is a skew symmetric matrix of odd order
Hence the determinant of the coefficient matrix is zero.
Therefore, the system of equations have non trivial solutions, and it has infinitely many solutions

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