If A = pmatrix 1 & 2 \\ 2 & -1 pmatrix, then which ONE of the following is A^8?

pmatrix 625 & 0 \\ 0 & 625 pmatrix

pmatrix 25 & 0 \\ 0 & 25 pmatrix

pmatrix 125 & 0 \\ 0 & 125 pmatrix

pmatrix 625 & 0 \\ 0 & 625 pmatrix

pmatrix 3125 & 0 \\ 0 & 3125 pmatrix

GATE CSE 2025 Session 2 Q11: Eighth power of a 2×2 symmetric matrix

If $A = \begin{pmatrix} 1 & 2 \\ 2 & -1 \end{pmatrix}$ , then which ONE of the following is $A^8$ ?

Try it: multiple choice: exactly one correct option

Wrong answer costs 1/3 mark in GATE.

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Answer: C (official GATE 2025 final answer key)

Concept used: Compute a small power first and look for a scalar multiple of the identity.

First step: $A^2 = 5I$ , so $A^8 = (A^2)^4 = 5^4 I$ .

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Source: official GATE 2025 Session 2 CS question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key