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GATE 2022 CE (CE1) – Question 25

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigenvalues and eigenvectors · 1 mark · Multiple select

$M=\begin{bmatrix}1&3\\4&2\end{bmatrix}$ has eigenvalues 5 and −2. For $Q=M^3-4M^2-2M$, which are eigenvalues of Q?

  1. 15
  2. 25
  3. −20
  4. −30

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Correct answer: (A) 15; (C) −20

Explanation

For any polynomial p, an eigenvector of M with eigenvalue λ has eigenvalue p(λ) for p(M). Here $p(\lambda)=\lambda^3-4\lambda^2-2\lambda$.

At λ=5 the result is $125-100-10=15$; at λ=−2 it is $-8-16+4=-20$. **Answer: A and C**.