Question

TRUE OR FALSE

Determine whether the statement “The matrix A is symmetric and matrix S is orthogonal, then matrix ${\mathit{S}}^{\mathbf{-}\mathbf{1}}\mathit{A}\mathit{S}$must be symmetric.

Short answer

The given statement is true.

Step-by-step solution

Definition of the symmetric matrix

“Matrix A is symmetric if $A^T=A$. Matrix B is orthogonal if and only if $B^T=B^{-1}$”

Explanation of the solution

Consider the statement as follows:

“The matrix A is symmetric and matrix S is orthogonal then the matrix $S^{-1}AS$is symmetric.

Consider the matrix as follows:

$Q=S^{-1}AS$

Taking transpose on both sides as follows.

$$\begin{gathered} Q=S^{-1}AS \\ {Q}^T={\left(S^{-1}AS\right)}^T \\ {Q}^T=S^T{A}^T{\left(S^{-1}\right)}^T \left(1\right) \end{gathered}$$

Since, A is symmetric matrix, then

$A^T=A$ ......(2)

SinceS is orthogonal as follows:

$$\begin{gathered} S^T=S^{-1} \\ {\left(S^{-1}\right)}^T=S \left(3\right) \end{gathered}$$

Using (2) and (3) in (1) as follows:

$$\begin{gathered} Q^T=S^{-1}AS \\ {Q}^T=Q \\ Q=S^{-1}AS \end{gathered}$$

Hence,$Q=S^{-1}AS$is symmetric.

Thus, the given statement “Matrix A is symmetric if $A^T=A$. Matrix B is orthogonal is and only if $B^T=B^{-1}$" is true.