Let $S=\left\{m \in Z: A^{m^{2}}+A^{m}=3 I-A^{-6}\right\}$, where $\mathrm{A}=\left[\begin{array}{cc}2 & -1 \\ 1 & 0\end{array}\right]$. Then $\mathrm{n}(S)$ is equal to ______.

JEE Main · 2025 · 29 Jan 2025 · Shift 1 · Matrices

Let

S = {m \in Z : A^{m^{2}} + A^{m} = 3 I - A^{- 6}}

, where

A = [21 - 1 0]

. Then

n (S)

is equal to ______.

Answer: 2

By Cayley–Hamilton,

A = (21 - 1 0)

satisfies

(A - I)^{2} = 0

, since its characteristic polynomial is

(λ - 1)^{2} = 0

. Writing

N = A - I

N^{2} = 0

, so for every integer

n

A^{n} = I + n N .

Then

A^{m^{2}} + A^{m} = 2 I + (m^{2} + m) N, 3 I - A^{- 6} = 3 I - (I - 6 N) = 2 I + 6 N .

Equating the coefficients of

N

m^{2} + m = 6 ⟹ m^{2} + m - 6 = 0 ⟹ (m + 3) (m - 2) = 0,

m = - 3

m = 2

. Hence

n (S) = 2

JEE Main · 2025 · 29 Jan 2025 · Shift 1 · Integer answer