Let $A$ be a $2 imes 2$ symmetric matrix such that $A\left[\begin{array}{l}1 \ 1\end{array} ight]=\left[\begin{array}{l}3 \ 7\end{array} ight]$ and the determinant of $A$ be 1 . If $A^{-1}=\alpha A+\beta I$, where $I$ is an identity matrix of order $2 imes 2$, then $\alpha+\beta$ equals…

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Let $A$ be a $2 	imes 2$ symmetric matrix such that $A\left[\begin{array}{l}1 \ 1\end{array}ight]=\left[\begin{array}{l}3 \ 7\end{array}ight]$ and the determinant of $A$ be 1 . If $A^{-1}=\alpha A+\beta I$, where $I$ is an identity matrix of order $2 	imes 2$, then $\alpha+\beta$ equals _______

Aicharya · Accepted Answer

$\begin{aligned} & 	ext { Let } A=\left[\begin{array}{ll}a & b \ b & d\end{array}ight] \ & {\left[\begin{array}{ll}a & b \ b & d\end{array}ight]\left[\begin{array}{l}1 \ 1\end{array}ight]=\left[\begin{array}{l}3 \ 7\end{array}ight], a d-b^2=1} \ & a+b=3, b+d=7,(3-b)(7-b)-b^2=1 \ & 21-10 b=1 ightarrow b=2, a=1, d=5 \ & A=\left[\begin{array}{ll}1 & 2 \ 2 & 5\end{array}ight], A^{-1}=\left[\begin{array}{cc}5 & -2 \ -2 & 1\end{array}ight] \ & A^{-1}=\alpha A+\beta I \ & {\left[\begin{array}{cc}5 & -2 \ -2 & 1\end{array}ight]=\left[\begin{array}{cc}\alpha+\beta & 2 \alpha \ 2 \alpha & 5 \alpha+\beta\end{array}ight]} \ & \alpha=-1, \beta=6 ightarrow \alpha+\beta=5\end{aligned}$

Let $A$ be a $2 \times 2$ symmetric matrix such that $A\left[\begin{array}{l}1 \\…

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