If the matrix A=1-1-11 then An+1=

for n=2, A3=A2·A =2-2-22 1-1-11 =4-4-44 =41-1-11=2n1-1-11

If the matrix A=1-1-11 then An+1=

for n=2, A3=A2·A =2-2-22 1-1-11 =4-4-44 =41-1-11=2n1-1-11

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If the matrix $A = (\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}) then A^{n + 1} =$

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$2 (\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix})$

$2 n (\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix})$

$2^{n} (\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix})$

$2^{n + 1} (\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix})$

answer is C.

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Detailed Solution

$for n = 2, A^{3} = A^{2} \cdot A = [\begin{matrix} 2 & - 2 \\ - 2 & 2 \end{matrix}] [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}] = [\begin{matrix} 4 & - 4 \\ - 4 & 4 \end{matrix}] = 4 [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}] = 2^{n} [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}]$