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GATE 2017 CE (CE1) – Question 38

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigenvalues and eigenvectors · 2 marks · Multiple choice

For the matrix $$A=\begin{pmatrix}5&-1\\4&1\end{pmatrix},$$ which statement about eigenvalues and eigenvectors is true?

  1. Eigenvalue 3 has multiplicity 2, with only one independent eigenvector.
  2. Eigenvalue 3 has multiplicity 2, with two independent eigenvectors.
  3. Eigenvalue 3 has multiplicity 2, with no independent eigenvectors.
  4. Eigenvalues are 3 and −3, with two independent eigenvectors.

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Correct answer: (A) Eigenvalue 3 has multiplicity 2, with only one independent eigenvector.

Explanation

The characteristic polynomial is $(5-\lambda)(1-\lambda)+4=\lambda^2-6\lambda+9=(\lambda-3)^2$. At $\lambda=3$, $A-3I$ has rows $(2,-1)$ and $(4,-2)$, so its rank is 1 and its null space has dimension 1. Thus the repeated eigenvalue has only one independent eigenvector, for example $(1,2)^T$. Select A.

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