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GATE 2026 IN – Question 11

Engineering Mathematics · Linear algebra and matrices · 1 mark · Multiple choice

A matrix $P\in\mathbb{R}^{3\times3}$ satisfies $P^3-4P^2+5P-2I=0$, where $I$ is a $3\times3$ identity matrix. The set of all possible eigenvalues of matrix $P$ is ________.

  1. $\{1,2,3\}$
  2. $\{1\}$
  3. $\{1,2\}$
  4. $\{2\}$

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Correct answer: (C) $\{1,2\}$

Explanation

Let $v\ne0$ be an eigenvector with eigenvalue $\lambda$, so $Pv=\lambda v$. Applying the given matrix polynomial to $v$ gives $(\lambda^3-4\lambda^2+5\lambda-2)v=0$.

Since $v\ne0$, $\lambda^3-4\lambda^2+5\lambda-2=(\lambda-1)^2(\lambda-2)=0$. Thus each eigenvalue is either 1 or 2; both values are possible. The correct set is $\{1,2\}$, **option C**.