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JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Matrices

Let

A = (1 - 2 2 - 5)

. Let

α, β \in R

be such that

α A^{2} + β A = 2 I

. Then

α + β

is equal to -

(A)-10
(B)-6
(C)6
(D)10✓ Correct

Answer: (D)

Solution

The characteristic equation of matrix

A

is

∣ A - λ I ∣ = 0

:

1 - λ - 2 2 - 5 - λ = 0

(1 - λ) (- 5 - λ) + 4 = 0

λ^{2} + 4 λ - 1 = 0

By Cayley-Hamilton theorem,

A^{2} + 4 A - I = 0

, so

A^{2} + 4 A = I

.

Multiplying by 2:

2 A^{2} + 8 A = 2 I

.

Comparing with

α A^{2} + β A = 2 I

, we get

α = 2

and

β = 8

.

Therefore

α + β = 10

. Answer: (D)

JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Single correct option

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