First slide

Operations of matrix

Question

Let $α$ be a root of the equation $x^{2} + x + 1 = 0$ and the matrix $A = \frac{1}{\sqrt{3}} [\begin{matrix} 1 & 1 & 1 \\ 1 & α & α^{2} \\ 1 & α^{2} & α^{4} \end{matrix}]$ , then the matrix $A^{31}$ is equal to

Easy

A

$I_{3}$
B

A
C

$A^{2}$
D

$A^{3}$

Solution

$A^{2} = \frac{1}{3} [\begin{matrix} 1 & 1 & 1 \\ 1 & ϖ & ϖ^{2} \\ 1 & ϖ^{2} & ϖ \end{matrix}] [\begin{matrix} 1 & 1 & 1 \\ 1 & ϖ & ϖ^{2} \\ 1 & ϖ^{2} & ϖ \end{matrix}] = [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}]$ , $(∵ 1 + ω + ω^{2} = 0)$
$\Rightarrow A^{4} = I$
$\Rightarrow A^{31} = A^{28} \times A^{3} = I \times A^{3} = A^{3}$

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