Let A be a symmetric matrix such that | A | = 2 and [ 2 1 3 3 2 ] A = [ 1 2 ] . If the sum of the diagonal elements of A is s , then s 2 is equal to ________ . [2023]

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Solution

Q.
$Let A be a symmetric matrix such that | A | = 2 and$ $[\begin{matrix} 2 & 1 \\ 3 & \frac{3}{2} \end{matrix}]$ $A = [\begin{matrix} 1 & 2 \\ α & β \end{matrix}] .$ If the sum of the diagonal elements of A is $s$ , then $\frac{β s}{α^{2}}$ is equal to ________ . [2023]

Ans.
(5)

Given, $| A | = 2$

Let $A = [\begin{matrix} a & b \\ b & c \end{matrix}] \Rightarrow a c - b^{2} = 2 ...(i)$

Also, $[\begin{matrix} 2 & 1 \\ 3 & \frac{3}{2} \end{matrix}] [\begin{matrix} a & b \\ b & c \end{matrix}] = [\begin{matrix} 1 & 2 \\ α & β \end{matrix}]$

$\Rightarrow [\begin{matrix} 2 a + b & 2 b + c \\ 3 a + \frac{3}{2} b & 3 b + \frac{3}{2} c \end{matrix}] = [\begin{matrix} 1 & 2 \\ α & β \end{matrix}]$

$\Rightarrow 2 a + b = 1 ...(ii), 2 b + c = 2 ...(iii)$

Solving (i), (ii), and (iii) we get $b = - \frac{1}{2}, a = \frac{3}{4}, c = 3$

Now, $α = 3 a + \frac{3}{2} b = \frac{3}{2}, β = 3 b + \frac{3}{2} c = 3$

$s = a + c = \frac{15}{4}$ $∴$ $\frac{β s}{α^{2}} = \frac{3 \times \frac{15}{4}}{\frac{9}{4}} = 5$

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