Let A=pmatrix1&2-1&4pmatrix.

Question

Let $A=\begin{pmatrix}1&2\-1&4\end{pmatrix}$. If $A^{-1}=\alpha I+\beta A$, $\alpha,\beta\in\mathbb{R}$, $I$ is a $2	imes2$ identity matrix, then $4(\alpha-\beta)$ is equal to:

Accepted Answer

$|A|=6$, $A^{-1}=\dfrac{\operatorname{adj}A}{|A|}=\dfrac16\begin{pmatrix}4&-2\1&1\end{pmatrix}=\begin{pmatrix}\frac23&-\frac13\\frac16&\frac16\end{pmatrix}$.

Set $\alpha I+\beta A=\begin{pmatrix}\alpha+\beta&2\beta\-\beta&\alpha+4\beta\end{pmatrix}$. Matching $\alpha+\beta=\dfrac23$ and $\beta=-\dfrac16$: $\alpha=\dfrac23+\dfrac16=\dfrac56$.

$4(\alpha-\beta)=4\left(\dfrac56+\dfrac16ight)=4(1)=4$.

Solutions · 1