GATE CS 2026 Set 1 Q.13: Linear Algebra

For $n > 1$ , the maximum multiplicity of any eigenvalue of an $n \times n$ matrix with elements from $\mathbb{R}$ is

$n$

$n-1$

$1$

$n+1$

Official answer and solution

Official answer: A
From: Official GATE 2026 answer key

Solution

Use the degree of the characteristic polynomial of an $n \times n$ matrix.

The characteristic polynomial $p(\lambda) = \det(A - \lambda I)$ of an $n \times n$ real matrix $A$ has degree $n$ . For example, the scalar matrix $A = cI_n$ has characteristic polynomial $(c - \lambda)^n$ and eigenspace $\mathbb{R}^n$ , so the eigenvalue $\lambda = c$ has both algebraic and geometric multiplicity $n$ . Thus the maximum multiplicity is $n$ .

Final answer

Option (A)

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

Source

Paper: GATE 2026 Computer Science and Information Technology - Session 1 (CS1)
Question: 13 of 65
Section: Computer Science and Information Technology

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