Let \(A\) be a \(2\times 2\) symmetric matrix such that \(A\begin{bmatrix}1\\1\end{bmatrix}=\begin{bmatrix}3\\7\end{bmatrix}\) and the determinant of \(A\) be 1. If \(A^{-1}=\alpha A+\beta I\), where \(I\) is the identity matrix of order \(2\times 2\), then \(\alpha+\beta\) equals ____.

JEE Main · 2024 · 04 Apr 2024 · Shift 2 · Matrices

Let

A

be a

2 \times 2

symmetric matrix such that

A [11] = [37]

and the determinant of

A

be 1. If

A^{- 1} = α A + β I

, where

I

is the identity matrix of order

2 \times 2

, then

α + β

equals ____.

Answer: 5

Let

A = [a c c b]

(symmetric).

A [11] = [37] \Rightarrow a + c = 3, c + b = 7

det A = ab - c^{2} = 1

.

From

a = 3 - c, b = 7 - c

(3 - c) (7 - c) - c^{2} = 1 \Rightarrow 21 - 10 c + c^{2} - c^{2} = 1 \Rightarrow 10 c = 20 \Rightarrow c = 2, a = 1, b = 5

A = [1225]

Characteristic equation (Cayley–Hamilton):

A^{2} - (tr A) A + (det A) I = 0 \Rightarrow A^{2} - 6 A + I = 0 \Rightarrow A^{- 1} = 6 I - A = - A + 6 I

.

So

α = - 1, β = 6 \Rightarrow α + β = 5

JEE Main · 2024 · 04 Apr 2024 · Shift 2 · Integer answer