Let A be a 2 2 symmetric matrix such that A [ l 1 1 ]= [ l 3 7 ] … - Vidyadip

Question

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

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Answer

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

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