Let A be a symmetric matrix such that \(|\mathrm{A}|=2\) and \(\left[\begin{array}{ll}2 & 1 \\ 3 & \frac{3}{2}\end{array}\right] \mathrm{A}=\left[\begin{array}{ll}1 & 2 \\ \alpha & \beta\end{array}\right]\). If the sum of the diagonal elements of A is s, then \(\frac{\beta \mathrm{s}}{\alpha^{2}}\) is equal to ______.

JEE Main · 2023 · 29 Jan 2023 · Shift 2 · Matrices

Let A be a symmetric matrix such that

∣ A ∣ = 2

and

[23 1 \frac{3}{2}] A = [1 α 2 β]

. If the sum of the diagonal elements of A is s, then

\frac{β s}{α ^{2}}

is equal to ______.

Answer: 5

Let

A = [a b b c]

(symmetric). The product

[23 1 \frac{3}{2}] A

gives

2 a + b = 1

and

2 b + c = 2

, so

a = \frac{1 - b}{2}

and

c = 2 - 2 b

. Then

∣ A ∣ = a c - b^{2} = (1 - b)^{2} - b^{2} = 1 - 2 b = 2

forces

b = - \frac{1}{2}

, whence

a = \frac{3}{4}

and

c = 3

. The remaining entries are

α = 3 a + \frac{3}{2} b = \frac{3}{2}

and

β = 3 b + \frac{3}{2} c = 3

, and the trace is

s = a + c = \frac{15}{4}

. Therefore

\frac{β s}{α ^{2}} = \frac{3 \cdot \frac{15}{4}}{( \frac{3}{2} ) ^{2}} = 5

JEE Main · 2023 · 29 Jan 2023 · Shift 2 · Integer answer