Question:
If $\mathrm{A}=\left(\begin{array}{ll}2 & 2 \\ 9 & 4\end{array}\right)$ and $\mathrm{I}=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$, then $10 \mathrm{~A}^{-1}$ is equal to:
Correct Option: , 3
Solution:
Characteristics equation of matrix ' $A$ ' is $|A-\lambda I|=0$
$\left|\begin{array}{cc}2-\lambda & 2 \\ 9 & 4-\lambda\end{array}\right|=0 \Rightarrow \lambda^{2}-6 \lambda-10=0$
$\therefore \quad A^{2}-6 A-10 I=0$
$\Rightarrow A^{-1}\left(A^{2}\right)-6 A^{-1}-10 I A^{-1}=0$
$\Rightarrow 10 A^{-1}=A-6 I$
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