GATE CS 2025 Set 2 | Question 11

MCQ · 1 mark · −0.33

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

  1. Option A: \(\begin{pmatrix} 25 & 0 \\ 0 & 25 \end{pmatrix}\)
  2. Option B: \(\begin{pmatrix} 125 & 0 \\ 0 & 125 \end{pmatrix}\)
  3. Option C: \(\begin{pmatrix} 625 & 0 \\ 0 & 625 \end{pmatrix}\) (correct)
  4. Option D: \(\begin{pmatrix} 3125 & 0 \\ 0 & 3125 \end{pmatrix}\)

Answer

(C)

Explanation

Square the matrix first. The diagonal entries are \(1 \cdot 1 + 2 \cdot 2 = 5\) and \(2 \cdot 2 + (-1)(-1) = 5\), and both off-diagonal entries are \(1 \cdot 2 + 2 \cdot (-1) = 0\), so

\[A^{2} = \begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix} = 5I.\]

Then \(A^{8} = (A^{2})^{4} = 5^{4}I = 625I\). The other options are \(5^{2}I\), \(5^{3}I\) and \(5^{5}I\), which are \(A^{4}\), \(A^{6}\) and \(A^{10}\).

Answer: C