JEE Main · 2024 · 27 Jan 2024 · Shift 2 · Matrices

Let A be a

2 \times 2

real matrix and I be the identity matrix of order 2 . If the roots of the equation

∣ A - x I ∣ = 0

be -1 and 3 , then the sum of the diagonal elements of the matrix

A^{2}

is ______

Answer: 10

Solution

The characteristic equation

∣ A - x I ∣ = 0

of a

2 \times 2

matrix

A

x^{2} - tr (A) x + det (A) = 0

. Its roots are

- 1

and

3

, so

tr (A) = (- 1) + 3 = 2, det (A) = (- 1) (3) = - 3.

By the Cayley–Hamilton theorem,

A

satisfies its own characteristic equation:

A^{2} - tr (A) A + det (A) I = 0 \Rightarrow A^{2} = 2 A - (- 3) I = 2 A + 3 I .

Taking the trace of both sides:

tr (A^{2}) = 2 tr (A) + 3 \cdot 2 = 2 (2) + 6 = 10.

JEE Main · 2024 · 27 Jan 2024 · Shift 2 · Integer answer