Question

If $A$ is an invertible matrix and $B$ is an orthogonal matrix, of the order same as that of $A$, then $C=$ $A^{-1} B A$ is

• A. an orthogonal matrix
• B. symmetric matrix
• C. skew-symmetric matrix
• D. none of these

Answer 1

Answer: (D)

Let $B=\begin{pmatrix}\cos (\pi / 2) & \sin (\pi / 2) \\ -\sin (\pi / 2) & \cos (\pi / 2)\end{pmatrix}=\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}$
and
$A=\begin{pmatrix}1 & 3 \\0 & 1\end{pmatrix}, A^{-1}=\begin{pmatrix}1 & -3 \\0 & 1\end{pmatrix}$
Note that $B$ is an orthogonal matrix.
$C =A^{-1} B A=\begin{pmatrix}1 & -3 \\0 & 1
\end{pmatrix}\begin{pmatrix}0 & 1 \\-1 & 0\end{pmatrix}\begin{pmatrix}1 & 3 \\0 & 1\end{pmatrix}$
$=\begin{pmatrix}3 & 10 \\-1 & -3\end{pmatrix}$
Note that $C$ is neither symmetric, nor skew-symmetric and nor-orthogonal.