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  4. If [1 /25&0 x&1 /25]=[5&0 -a&5]-2 then the value of x is

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Q. If $\begin{bmatrix}1 /25&0\\ x&1 /25\end{bmatrix}=\begin{bmatrix}5&0\\ -a&5\end{bmatrix}^{-2}$ then the value of $x$ is

Matrices

Solution:

Let $A=\begin{bmatrix}5&0\\ -a&5\end{bmatrix}\Rightarrow adj \left(A\right)=\begin{bmatrix}5&0\\ a&5\end{bmatrix}\Rightarrow A^{-1} $
$=\frac{1}{\left|A\right|}\begin{bmatrix}5&0\\ a&5\end{bmatrix}=\frac{1}{25}\begin{bmatrix}5&0\\ a&5\end{bmatrix}$
$\Rightarrow A^{-2} =\left(A^{-1}\right)^{2} =\frac{1}{25}\begin{bmatrix}5&0\\ a&5\end{bmatrix}\frac{1}{25}\begin{bmatrix}5&0\\ a&5\end{bmatrix}$
$=\frac{1}{625}\begin{bmatrix}25&0\\ 10a&25\end{bmatrix}\begin{bmatrix}\frac{1}{25}&0\\ \frac{2a}{125}&\frac{1}{25}\end{bmatrix}$
Now $\begin{bmatrix}1 /25&0\\ x&1 25\end{bmatrix} = \begin{bmatrix}\frac{1}{25}&0\\ \frac{2a}{125}&\frac{1}{25}\end{bmatrix} \Rightarrow x = 2a /125$