First slide

Transpose of matrix

Question

If A and B are symmetric matrices, then AB- BA is a

Moderate

A

symmetric matrix
B

skew-symmetric matrix
C

diagonal matrix
D

null matrix

Solution

We are given $A^{'} = A, B^{'} = B$
now,
$\begin{matrix} (AB - BA)^{'} = (AB)^{'} - (BA)^{'} = B^{'} A^{'} - A^{'} B^{'} \\ = BA - AB = - (AB - BA) \\ (AB - BA)^{'} = - (AB - BA) \end{matrix}$
Hence, AB - BA is a skew-symmetric matrix.

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