Question

If the$\mathbf{n}\mathbf{\times }\mathbf{n}$ matrices Aand Bare orthogonal, which of the matrices in Exercise 5 through 11 must be orthogonal as well?${\mathbf{B}}^{\mathbf{-}\mathbf{1}}$.

Short answer

The Matrix${\mathrm{B}}^{-1}$ is orthogonal.

Step-by-step solution

Definition of Orthogonal transformation

An$\mathbf{n}\mathbf{\times }\mathbf{n}$matrix Ais orthogonal if and only if its columns form an orthonormal basis of${\mathbf{ℝ}}^{\mathbf{n}}$.

Verification whether the given matrix is orthogonal.

Given thatA andB are orthogonal matrices.

By theorem:

The inverse$A^{-1}$ of an orthogonal$n\times n$ matrixA is orthogonal.

Therefore,$B^{-1}$ is an orthogonal matrix.