Question

TRUE OR FALSE?If matrices A and Sare orthogonal, then ${\mathbf{S}}^{\mathbf{-}\mathbf{1}}\mathbf{}\mathbf{AS}$is orthogonal as well.

Short answer

The given statement is true.

Step-by-step solution

Properties of orthogonal matrices.

According to the properties of orthogonal matrices, the product of two orthogonal $\mathbf{n}\mathbf{\times }\mathbf{n}$matrices is also orthogonal and the inverse of an$\mathbf{n}\mathbf{\times }\mathbf{n}$matrix is also orthogonal.

Check whether the given statement is a true or false.

The given statement is if matrices A and S are orthogonal, then $S^{-1}AS$must is orthogonal as well.

It is given that A and S are orthogonal.

Here, S is orthogonal, so, $S^{-1}$is also orthogonal.

This means, $S^{-1}AS$is also orthogonal.

Then the given statement is true, by the properties of orthogonal matrices.