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

Question

Let $A$ be a $2	imes 2$ symmetric matrix such that $A[\begin{gathered} 1 \ 1 \end{gathered}]=[\begin{gathered} 3 \ 7 \end{gathered}]$ 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 \_\_\_\_\_\_\_\_\_.

Answer: 5

Accepted Answer

Let $A=[\begin{gathered} a b \ b c ]$

$|A|=1\Rightarrow ac-{b}^{2}=0\,$

$[ a b \ b c ][\begin{gathered} 1 \ 1 \end{gathered}]=[\begin{gathered} 3 \ 7 \end{gathered}]$

$\begin{gathered} \Rightarrow \, a+b=3\, \ b+c=7\, \end{gathered}$

from (i), (ii) and (iii) $a=1,b=2,c=5$

$\begin{gathered} \Rightarrow \quad A=[1 \quad 2 \ 2 \quad 5]\Rightarrow {A}^{-1}=[5 \quad -2 \ -2 \quad 1] \ {A}^{-1}=\alpha A+\beta I \ \Rightarrow \quad [5 \quad -2 \ -2 \quad 1]=\alpha [1 \quad 2 \ 2 \quad 5]+\beta [1 \quad 0 \ 0 \quad 1 \end{gathered}] \ \Rightarrow \quad  \alpha =-1\beta =6 \ \alpha +\beta =5 \end{gathered}$

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