Matrices & Determinants
Symmetric matrix, determinant, system of equations
nta_pyq_2023_jan
Grade 12
Question:
Let A be a symmetric matrix such that $|A| = 2$ and $\begin{pmatrix} 2 & 1 \\ 3 & \frac{3}{2} \end{pmatrix} A = \begin{pmatrix} 1 & 2 \\ \alpha & \beta \end{pmatrix}$. If the sum of the diagonal elements of A is $s$, then $\frac{\beta s}{\alpha^2}$ is equal to ___.
Step-by-Step Solution
Key Concept: Use the given matrix equation to find the entries of A, apply symmetry constraint and $|A|=2$
Let $A = \begin{pmatrix}a & b \\ b & c\end{pmatrix}$ (symmetric). From $\begin{pmatrix}2&1\\3&3/2\end{pmatrix}\begin{pmatrix}a&b\\b&c\end{pmatrix} = \begin{pmatrix}1&2\\\alpha&\beta\end{pmatrix}$: $2a+b=1$, $2b+c=2$, $ac-b^2=2$. Solving: $b=-1/2$, $a=3/4$, $c=3$. $\alpha = 3a+\frac{3b}{2} = 9/4 - 3/4 = 3/2$, $\beta = 3b + \frac{3c}{2} = -3/2 + 9/2 = 3$. $s = a+c = 3/4+3 = 15/4$. $\frac{\beta s}{\alpha^2} = \frac{3 \times 15/4}{9/4} = 5$. Answer: 5
Correct Answer: 5