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(A) Given $A = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix}$.Calculate powers of $A$:$A^2 = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \begin{bmatri

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$32$

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(A) Given $A = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix}$.Calculate powers of $A$:$A^2 = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 2 \ 2 & 2 \end{bmatrix} = 2^1 A$.$A^3 = A^2 \cdot A = \begin{bmatrix} 2 & 2 \ 2 & 2 \end{bmatrix} \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} = \begin{bmatrix} 4 & 4 \ 4 & 4 \end{bmatrix} = 2^2 A$.By induction,$A^n = 2^{n-1} A$.For $n = 6$,$A^6 = 2^{6-1} A = 2^5 A = 32 A$.Comparing $A^6 = k A$ with $A^6 = 32 A$,we get $k = 32$.

If a matrix $A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$ satisfies $A^6 = k A$,then the value of $k$ is

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