First slide

Operations of matrix

Question

If A = $A = [\begin{matrix} i & - i \\ - i & i \end{matrix}] a n d B = [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}]$ the A⁸ equals

Moderate

A

64 B
B

-64 B
C

-128 B
D

128 B

Solution

We have,
$A = [\begin{matrix} i & - i \\ - i & i \end{matrix}] = i [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}] = i B$
$\Rightarrow A^{2} = {(i B)}^{2} = i^{2} B^{2} = - B^{2} = - [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}] [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}]$
$= [\begin{matrix} - 2 & 2 \\ 2 & - 2 \end{matrix}] = - 2 B$
⇒ A⁴ = (–2B)²= 4^B2 = 4(2B) = 8B
⇒ A⁸ = (A⁴)² = (8B)² = 64B² = 128B

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