Matrices Question 9
Question 9 - 29 January - Shift 2
Let $A$ be a symmetric matrix such that $|A|=2$ and $ \begin{bmatrix} 2 & 1 \\ 3 & \frac{3}{2}\end{bmatrix} A= \begin{bmatrix} 1 & 2 \\ \alpha & \beta\end{bmatrix} $. If the sum of the diagonal elements of $A$ is $s$, then $\frac{\beta s}{\alpha^{2}}$ is equal to ______
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Answer: 5
Solution:
Formula: Properties of Matrix Multiplication
$ \begin{bmatrix} 2 & 1 \\ 3 & \frac{3}{2} \end{bmatrix} \begin{bmatrix} a & b \\ b & c \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ \alpha & \beta \end{bmatrix} $
Now $a c-b^{2}=2$ and $2 a+b=1$ and $2 b+c=2$
Solving all these above equations we get
$\frac{1-b}{2} \times(\frac{2-2 b}{1})-b^{2}=2$
$\Rightarrow(1-b)^{2}-b^{2}=2$
$\Rightarrow 1-2 b=2$
$\Rightarrow b=-\frac{1}{2}$ and $a=\frac{3}{4}$ and $c=3$
Hence $\alpha=3 a+\frac{3 b}{2}=\frac{9}{4}-\frac{3}{4}=\frac{3}{2}$
and $\beta=3 b+\frac{3 c}{2}=-\frac{3}{2}+\frac{9}{2}=3$
also $s=a+c=\frac{15}{4}$
$\therefore \frac{\beta s}{\alpha^{2}}=\frac{3 \times 15}{4 \times \frac{9}{4}}=5$
