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  4. Let A and B be any two 3 × 3 matrices. If A is symmetric and B is skew symmetric, then the matrix AB - BA is:

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Q. Let A and B be any two 3 × 3 matrices. If A is symmetric and B is skew symmetric, then the matrix AB - BA is:

Matrices

Solution:

Let A be symmetric matrix and B be skew symmetric matrix.
$\therefore \, A^T = A$ and $B^T = -B$
Consider
$(AB - BA)^T = (AB)^T - (BA)^T$
$= B^TA^T - A^TB^T$
$= (-B) (A) - (A) (-B)$
$= -BA + AB = AB - BA$
This shows AB - BA is symmetric matrix.