If A=pmatrix2&29&4pmatrix and I=pmatrix1&00&1pmatrix, then 10A^-1 is equal to:

Question

If $A=\begin{pmatrix}2&2\9&4\end{pmatrix}$ and $I=\begin{pmatrix}1&0\0&1\end{pmatrix}$, then $10A^{-1}$ is equal to:

Accepted Answer

The characteristic equation of $A$: $\lambda^2-6\lambda-10=0$, so $A^2-6A-10I=0 \Rightarrow A^2=6A+10I$.

Multiplying by $A^{-1}$: $A-6I-10A^{-1}=0 \Rightarrow 10A^{-1}=A-6I$.

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