If the matrix A=[1−1−11]then An+1

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If the matrix $A = [\begin{array}{cc} 1 & - 1 \\ - 1 & 1 \end{array}]$ then $A^{n + 1}$

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$2^{n + 1} [\begin{array}{cc} 1 & - 1 \\ - 1 & 1 \end{array}]$

$2^{n} [\begin{array}{cc} 1 & - 1 \\ - 1 & 1 \end{array}]$

$n [\begin{array}{cc} 1 & - 1 \\ - 1 & 1 \end{array}]$

answer is C.

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Detailed Solution

$A ² =$ $[\begin{array}{cc} 1 & - 1 \\ - 1 & 1 \end{array}] [\begin{array}{cc} 1 & - 1 \\ - 1 & 1 \end{array}] = [\begin{array}{cc} 2 & - 2 \\ - 2 & 2 \end{array}] = 2 [\begin{array}{cc} 1 & - 1 \\ - 1 & 1 \end{array}] = 2$

$A^{3} = A^{2} A = 2 A \cdot A = 2 A^{2} = 2 (2 A) = 2^{2} A$

$∴ A^{n + 1} = 2^{n} A = 2^{n} [\begin{array}{cc} 1 & - 1 \\ - 1 & 1 \end{array}]$

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