# Linear Algebra I: Orthogonal Matrix Condition

1. Dec 10, 2011

I would like to check my reasoning for this problem to make sure I understand what an orthogonal matrix is.

1. The problem statement, all variables and given/known data

Determine if the matrix is orthogonal. If orthogonal, find the inverse.

$$\begin{pmatrix} -1 & 2 & 2\\ 2 & -1 & 2\\ 2 & 2 & -1 \end{pmatrix}$$

2. Relevant equations

If a matrix $A$ is orthogonal, then
$$A^{-1} = A^T.$$

3. The attempt at a solution

One of the conditions that must be met for a matrix to be orthogonal is that the length of the vectors spanning its column space must be 1, correct? So, if we let
$$A=\begin{pmatrix} -1 & 2 & 2\\ 2 & -1 & 2\\ 2 & 2 & -1 \end{pmatrix}=(\mathbf{a}_1,\mathbf{a}_2,\mathbf{a}_3),$$
then
$$||\mathbf{a}_1||^2 \neq 1,$$
so a condition for orthogonality has been violated. Thus, $A$ is not orthogonal and there is no need to continue with the problem.

Another way to put it would be to say that the $Col\ A$ is not an orthonormal set, so $A$ is not orthogonal. Is this correct?

2. Dec 10, 2011

### micromass

Staff Emeritus
That is all correct!!

3. Dec 10, 2011