Question

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

• 2 & 2 & - 1 \end{bmatrix}$is

• A. Orthogonal
• B. Involutary
• C. Idempotent
• D. NilpotentH

Answer 1

Answer: (A)

Orthogonal

Answer 2

Since for given $A = \frac{1}{3}\left| \begin{matrix} 1 & 2 & 2 \ 2 & 1 & - 2 \

• 2 & 2 & - 1 \end{matrix} \right|$. For orthogonal matrix

$AA^{T} = A^{T}A = I_{(3 \times 3)}$

⇒$AA^{T} = \frac{1}{3}\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}{3}\begin{bmatrix} 9 & 0 & 0 \ 0 & 9 & 0 \ 0 & 0 & 9 \end{bmatrix} = 3I$. Similarly$\alpha = 5$. Hence A is orthogonal