Question

TRUE OR FALSE?If matrix A is orthogonal, then the matrix${\mathit{A}}^{\mathbf{T}}$ must be orthogonal as well.

Short answer

The given statement is true.

Step-by-step solution

Recollect the properties of the orthogonal a matrix.

According to the properties of a orthogonal matrix, if a$\mathit{n}\mathbf{\times }\mathit{n}$matrix Ais an orthogonal matrix, then${\mathit{A}}^{\mathbf{-}\mathbf{1}}\mathbf{=}{\mathit{A}}^{\mathbf{T}}$.

Also, if Ais an orthogonal matrix, then${\mathit{A}}^{\mathbf{-}\mathbf{1}}$is also orthogonal.

Check whether the given statement is a true or false.

The given statement is if the matrix A is orthogonal, then the matrix$A^T$ must be orthogonal as well.

Here,A is orthogonal. This means,$A^{-1}$ is also orthogonal, and$A^{-1}=A^T$

This means,$A^T$ is also an orthogonal matrix.

It is clear that, the given statement is true, by the properties of orthogonal matrices.