GATE CS 2025 Set 2 Q.11: Linear Algebra

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

$\begin{pmatrix} 25 & 0 \\ 0 & 25 \end{pmatrix}$

$\begin{pmatrix} 125 & 0 \\ 0 & 125 \end{pmatrix}$

$\begin{pmatrix} 625 & 0 \\ 0 & 625 \end{pmatrix}$

$\begin{pmatrix} 3125 & 0 \\ 0 & 3125 \end{pmatrix}$

Official answer and solution

Solution

Compute $A^2$ first, observe that $A^2 = 5I$ , and raise it to the 4th power.

Computing $A^2$ gives:

A^2 = \begin{pmatrix} 1 & 2 \\ 2 & -1 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 2 & -1 \end{pmatrix} = \begin{pmatrix} 1 + 4 & 2 - 2 \\ 2 - 2 & 4 + 1 \end{pmatrix} = \begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix} = 5I

Therefore:

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

Final answer

Option (C)

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Paper: GATE 2025 Computer Science and Information Technology - Session 2 (CS2)
Question: 11 of 65
Section: Computer Science and Information Technology

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