First slide

Operations of matrix

Question

If $A = [\begin{matrix} 1 & \frac{1}{2} \\ 0 & 1 \end{matrix}] t h e n A^{64} i s$

Moderate

A

$[\begin{matrix} 1 & 32 \\ 32 & 1 \end{matrix}]$
B

$[\begin{matrix} 1 & 0 \\ 32 & 1 \end{matrix}]$
C

$[\begin{matrix} 1 & 32 \\ 0 & 1 \end{matrix}]$
D

None of these

Solution

$A^{2} = [\begin{matrix} 1 & \frac{1}{2} \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & \frac{1}{2} \\ 0 & 1 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$
$A^{4} = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] = [\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}]$
$A^{8} = [\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}] = [\begin{matrix} 1 & 4 \\ 0 & 1 \end{matrix}]$
Similarly, $A^{64} = [\begin{matrix} 1 & 32 \\ 0 & 1 \end{matrix}]$

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