If a matrix \(A=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]\) satisfies \(A^6=k A^{\prime}\), then the value of \(k\) is

\(A=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]\)
\(\begin{aligned} & A^2=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]=\left[\begin{array}{ll}2 & 2 \\ 2 & 2\end{array}\right] \\ & A^3=\left[\begin{array}{ll}2 & 2 \\ 2 & 2\end{array}\right]\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]=\left[\begin{array}{ll}4 & 4 \\ 4 & 4\end{array}\right]=\left[\begin{array}{ll}2^2 & 2^2 \\ 2^2 & 2^2\end{array}\right] \\ & A^4=\left[\begin{array}{ll}8 & 8 \\ 8 & 8\end{array}\right]=\left[\begin{array}{ll}2^3 & 2^3 \\ 2^3 & 2^3\end{array}\right] \\ & A^6=\left[\begin{array}{ll}2^5 & 2^5 \\ 2^5 & 2^5\end{array}\right]=2^5\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right] \Rightarrow 2^5=32 \\ & \therefore \mathrm{k}=32\end{aligned}\)