First slide

Transpose of matrix

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

Moderate

A

an orthogonal matrix
B

symmetric matrix
C

skew-symmetric matrix
D

none of these

Solution

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

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