The matrix A = frac{1}{3}begin{bmatrix} 1 & 2 | vedclass

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(A) matrix $A$ is said to be orthogonal if $AA^T = A^TA = I$,where $I$ is the identity matrix.Given $A = \frac{1}{3}\begin{bmatrix} 1 & 2 & 2 \ 2 & 1

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Orthogonal

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(A) matrix $A$ is said to be orthogonal if $AA^T = A^TA = I$,where $I$ is the identity matrix.Given $A = \frac{1}{3}\begin{bmatrix} 1 & 2 & 2 \ 2 & 1 & -2 \ -2 & 2 & -1 \end{bmatrix}$.Then $A^T = \frac{1}{3}\begin{bmatrix} 1 & 2 & -2 \ 2 & 1 & 2 \ 2 & -2 & -1 \end{bmatrix}$.Now,$AA^T = \frac{1}{9} \begin{bmatrix} 1 & 2 & 2 \ 2 & 1 & -2 \ -2 & 2 & -1 \end{bmatrix} \begin{bmatrix} 1 & 2 & -2 \ 2 & 1 & 2 \ 2 & -2 & -1 \end{bmatrix}$$= \frac{1}{9} \begin{bmatrix} 1+4+4 & 2+2-4 & -2+4-2 \ 2+2-4 & 4+1+4 & -4+2+2 \ -2+4-2 & -4+2+2 & 4+4+1 \end{bmatrix}$$= \frac{1}{9} \begin{bmatrix} 9 & 0 & 0 \ 0 & 9 & 0 \ 0 & 0 & 9 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} = I$.Since $AA^T = I$,the matrix $A$ is an orthogonal matrix.

The matrix $A = \frac{1}{3}\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ -2 & 2 & -1 \end{bmatrix}$ is

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