If 1&-1 -1&1 , satisfies the matrix equation A^2 = kA, write the … - Vidyadip

Question

If $\begin{bmatrix}1&-1\\-1&1\end{bmatrix},$ satisfies the matrix equation $A^2 = kA$, write the value of $k$.

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Answer

Given,
$\text{A}=\begin{bmatrix}1&-1\\-1&1\end{bmatrix}$
and
$\text{A}^2=\text{kA}$
$\Rightarrow\begin{bmatrix}1&-1\\-1&1\end{bmatrix}\begin{bmatrix}1&-1\\-1&1\end{bmatrix}=\text{k}\begin{bmatrix}1&-1\\-1&1\end{bmatrix}$
$\Rightarrow\begin{bmatrix}1+1&-1-1\\-1-1&1+1\end{bmatrix}=\begin{bmatrix}\text{k}&-\text{k}\\-1&1\end{bmatrix}$
$\Rightarrow\begin{bmatrix}2&-2\\-2&2\end{bmatrix}=\begin{bmatrix}\text{k}&-\text{k}\\-\text{k}&\text{k}\end{bmatrix}$
Since, corresponding entries of equal matrices are equal, so
$k = 2$

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