The GATE Grind

GATE 2026 CS (CS1) – Question 13

Engineering Mathematics · Linear Algebra · 1 mark · Multiple choice

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

  1. $n$
  2. $n - 1$
  3. 1
  4. $n + 1$

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Correct answer: (A) $n$

Explanation

The characteristic polynomial of an $n \times n$ matrix $A$ is given by $p(\lambda) = \det(A - \lambda I)$, which is a polynomial of degree $n$ in $\lambda$. By the Fundamental Theorem of Algebra, a degree $n$ polynomial has at most $n$ roots counting algebraic multiplicities.

For example, the identity matrix $I_{n \times n}$ has characteristic polynomial $(1 - \lambda)^n = 0$, where the eigenvalue $\lambda = 1$ has algebraic multiplicity $n$ (and geometric multiplicity $n$).

Thus, the maximum possible multiplicity of an eigenvalue of an $n \times n$ matrix is $n$. Therefore, option (A) is correct.