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  4. If A = [1&1 1&1] and A100 = 2kA, then the value of k is

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Q. If $A = \begin{bmatrix}1&1\\ 1&1\end{bmatrix} $ and $ A^{100} = 2^{k}A, $ then the value of $k$ is

Matrices

Solution:

Let $A = \begin{bmatrix}1&1\\ 1&1\end{bmatrix} $
$ A^{2} = \begin{bmatrix}1&1\\ 1&1\end{bmatrix}\begin{bmatrix}1&1\\ 1&1\end{bmatrix} = 2\begin{bmatrix}1&1\\ 1&1\end{bmatrix} = 2A $
$ A^{3} = 2^{2}\begin{bmatrix}1&1\\ 1&1\end{bmatrix}, A^{4} = 2^{3} \begin{bmatrix}1&1\\ 1&1\end{bmatrix}$
$A^{3} = 2^{2} A, A^{4} = 2^{3}A$
$ \therefore A^{n} = 2^{n-1}\begin{bmatrix}1&1\\ 1&1\end{bmatrix} $
$ \Rightarrow A^{100} = 2^{100-1} A $
$\Rightarrow A^{100} = 2^{99}A $
$ \Rightarrow 2^{k} A = 2^{99}A$
$ \Rightarrow k = 99 $