First slide

Operations of matrix

Question

If $A = [\begin{matrix} i & 0 \\ 0 & i \end{matrix}], n \in N$ , then A⁴ⁿ equals

Moderate

A

$[\begin{matrix} 0 & i \\ i & 0 \end{matrix}]$
B

$[\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$
C

$[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$
D

$[\begin{matrix} 0 & i \\ 0 & i \end{matrix}]$

Solution

We have,
$A^{2} = [\begin{matrix} i & 0 \\ 0 & i \end{matrix}] [\begin{matrix} i & 0 \\ 0 & i \end{matrix}] = [\begin{matrix} i^{2} & 0 \\ 0 & i^{2} \end{matrix}] = [\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}]$
$A^{4} = [\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}] [\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$
$∴ A^{4 n} = \frac{[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] . . . . . [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]}{n t i m e s} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

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