Question

If A and B are square matrices of same order and B is a skew-symmetric matrix, show that $A^{\prime }BA$ is a skew-symmetric matrix.

Answer 1

Given, A and B are square matrices of same order and B is a skew-symmetric matrix.

$B^{\prime }=-B$ .........(1)

Now, we have to prove that $A^{\prime }BA$ is a skew-symmetric matrix,

Consider, $\left[A^{\prime }\right(BA){]}^{\prime }=(BA{)}^{\prime }(A^{\prime }{)}^{\prime }$

$=A^{\prime }{B}^{\prime }A$

$=-A^{\prime }BA$ [From eq. 1]

Hence, $A^{\prime }BA$ is a skew-symmetric matrix.