If matrix $A=\left[\begin{array}{ll}1 & 2 \ 3 & 5\end{array} ight]$ and $A^{-1}=\alpha I+\beta A$ where $I$ is a unit matrix of order 2 and $\alpha, \beta$ are constants, then the value of $\alpha+\beta+\alpha \beta$ is

Question

If matrix $A=\left[\begin{array}{ll}1 & 2 \ 3 & 5\end{array}ight]$ and $A^{-1}=\alpha I+\beta A$ where $I$ is a unit matrix of order 2 and $\alpha, \beta$ are constants, then the value of $\alpha+\beta+\alpha \beta$ is

Aicharya · Accepted Answer

$\begin{aligned} & A^{-1}=\alpha I+\beta A \ & \Rightarrow\left[\begin{array}{ll}1 & 2 \ 3 & 5\end{array}ight]^{-1}=\alpha\left[\begin{array}{ll}1 & 0 \ 0 & 1\end{array}ight]+\beta\left[\begin{array}{ll}1 & 2 \ 3 & 5\end{array}ight] \ & \Rightarrow\left[\begin{array}{cc}-5 & 2 \ 3 & -1\end{array}ight]=\left[\begin{array}{cc}\alpha+\beta & 2 \beta \ 3 \beta & \alpha+5 \beta\end{array}ight] \ & \Rightarrow \alpha=-6, \beta=1\end{aligned}$ Now $\alpha+\beta+\alpha \beta=-6+1+(-6) 	imes 1=-11$

If matrix $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 5\end{array}\right]$ and $A^{-1}=\alpha I+\beta A$ where…

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