I would like to check my reasoning for this problem to make sure I understand what an orthogonal matrix is.(adsbygoogle = window.adsbygoogle || []).push({});

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

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

[tex]

\begin{pmatrix}

-1 & 2 & 2\\

2 & -1 & 2\\

2 & 2 & -1

\end{pmatrix}

[/tex]

2. Relevant equations

If a matrix [itex]A[/itex] is orthogonal, then

[tex]

A^{-1} = A^T.

[/tex]

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

[tex]

A=\begin{pmatrix}

-1 & 2 & 2\\

2 & -1 & 2\\

2 & 2 & -1

\end{pmatrix}=(\mathbf{a}_1,\mathbf{a}_2,\mathbf{a}_3),

[/tex]

then

[tex]

||\mathbf{a}_1||^2 \neq 1,

[/tex]

so a condition for orthogonality has been violated. Thus, [itex]A[/itex] isnotorthogonal and there is no need to continue with the problem.

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

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# Homework Help: Linear Algebra I: Orthogonal Matrix Condition

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