GATE 2025 CS (CS1) – Question 41

Engineering Mathematics · Linear Algebra · 2 marks · Multiple choice

Let $A$ be a $2 \times 2$ matrix as given.

$A = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$

What are the eigenvalues of the matrix $A^{13}$ ?

$1,\ -1$
$2\sqrt{2},\ -2\sqrt{2}$
$4\sqrt{2},\ -4\sqrt{2}$
$64\sqrt{2},\ -64\sqrt{2}$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (D) $64\sqrt{2},\ -64\sqrt{2}$

Explanation

The characteristic equation is λ^2 - 2 = 0, so λ = ±√2. The eigenvalues of A^13 are (±√2)^13 = ±2^6·√2 = ±64√2.