Question

If A and B are two skew-symmetric matrices of same order, then AB is symmetric if ______________.

Answer 1

It is given that, A and B are two skew-symmetric matrices of same order.

∴ A^T = −A and B^T = −B .....(1)

Now, the matrix AB is symmetric if

${\left(AB\right)}^T=AB$ (A matrix X is symmetric if X^T = X)

$\Rightarrow B^T{A}^T=AB$

$\Rightarrow \left(-B\right)\left(-A\right)=AB$ [Using (1)]

$\Rightarrow BA=AB$

Thus, if A and B are two skew-symmetric matrices of same order, then AB is symmetric if AB = BA.

If A and B are two skew-symmetric matrices of same order, then AB is symmetric if ___AB = BA___.