Let $A$ be a symmetric matrix such that $|A|=2$ and $\begin{bmatrix} 2 & 1 \[0.3em] 3 & \frac 32 \[0.3em] \end{bmatrix}𝐴−\begin{bmatrix} 1 & 2 \[0.3em] α & β \[0.3em] \end{bmatrix}$ . If the sum of the diagonal elements of $A$ is $s$ , then $\frac {βs}{α^2}$ is equal to ____ .

Question

Let $A$  be a symmetric matrix such that  $|A|=2$  and  $\begin{bmatrix} 2 & 1 \[0.3em] 3 & \frac 32 \[0.3em] \end{bmatrix}𝐴−\begin{bmatrix} 1 & 2 \[0.3em] α & β \[0.3em] \end{bmatrix}$ . If the sum of the diagonal elements of  $A$  is  $s$ , then  $\frac {βs}{α^2}$  is equal to ____ .

cdquestions Admin · Accepted Answer

(A)Let $ A = \begin{bmatrix} a & b b & c \end{bmatrix} $. Since $ |A| = 2 $: $$ ac - b^2 = 2. $$ (B)From the given equation: $$ \begin{bmatrix} 3 & -2  2 & 1 \end{bmatrix} \begin{bmatrix} a & b b & c \end{bmatrix} = \begin{bmatrix} 1 & 2 2 & 7 \end{bmatrix}. $$ Expanding row-wise gives equations: $$ 3a - 2b = 1, \quad 3b - 2c = 2, \quad 2a + b = 2, \quad 2b + c = 7. $$ (C)Solve these equations to find: $$ a = \frac{3}{4}, \, b = \frac{5}{4}, \, c = \frac{9}{2}. $$ Sum of diagonal elements: $$ s = a + c = \frac{3}{4} + \frac{9}{2} = \frac{21}{4}. $$ (D)Given $ \alpha = 3 $ and $ \beta = 15 $, compute: $$ \frac{\beta s}{\alpha^2} = \frac{15 \times \frac{21}{4}}{9} = 5. $$

Let \(A\) be a symmetric matrix such that \(|A|=2\) and \(\begin{bmatrix} 2 & 1 \\[0.3em] 3 & \frac 32 \\[0.3em] \end{bmatrix}𝐴−\begin{bmatrix} 1 & 2 \\[0.3em] α & β \\[0.3em] \end{bmatrix}\). If the sum of the diagonal elements of \(A\) is \(s\), then \(\frac {βs}{α^2}\) is equal to ____ .

Correct Answer: 5

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