If A = 1 & 2 2 & -1 , then which ONE of the following is A8 ? A) 25 & 0 0 & 25 B) 125 & 0 0 & 125 C) 625 & 0 0 & 625 D) 3125 & 0 0 & 3125

Official answer: C) 625 & 0 0 & 625. Explanation: Given the matrix: A = 1 & 2 2 & -1 Compute A2: A2 = 1 & 2 2 & -1 1 & 2 2 & -1 = 1(1) + 2(2) & 1(2) + 2(-1) 2(1) + (-1)(2) & 2(2) + (-1)(-1) = 5 & 0 0 & 5 = 5I Now, raise A2 to the fourth power to find A8: A8 = (A2)4 = (5I)4 = 54 I = 625 1 & 0 0 & 1 = 625 & 0 0 & 625 Thus, option (C) is the correct choice.

Linear Algebra

GATE CSE 2025 Set 2 · Q11

MCQ1 markEasy

Time00:00

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

Select one answerSingle correct

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Solution

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Official answer: C) 625 & 0 0 & 625

Linear Algebra

Given the matrix:

A = \begin{pmatrix} 1 & 2 \\ 2 & -1 \end{pmatrix}

Compute $A^2$ :

A^2 = \begin{pmatrix} 1 & 2 \\ 2 & -1 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 2 & -1 \end{pmatrix} = \begin{pmatrix} 1(1) + 2(2) & 1(2) + 2(-1) \\ 2(1) + (-1)(2) & 2(2) + (-1)(-1) \end{pmatrix} = \begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix} = 5I

Now, raise $A^2$ to the fourth power to find $A^8$ :

A^8 = (A^2)^4 = (5I)^4 = 5^4 I = 625 \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 625 & 0 \\ 0 & 625 \end{pmatrix}

Thus, option (C) is the correct choice.

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This Linear Algebra previous year question is Q11 of the GATE CSE 2025 Set 2 paper. Find more GATE CSE PYQs on Linear Algebra in Engineering Mathematics, or browse all GATE CSE PYQs.

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Linear Algebra in GATE CSE

Linear Algebra has 46 GATE CSE PYQs across 26 papers from 2011 to 2026. 13 of them are 1-mark MCQs like this one, and GATE CSE 2025 Set 2 has 3.

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Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.