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GATE 2022 BT – Question 12

Engineering Mathematics · Linear Algebra: Matrices and determinants, systems of linear equations, eigenvalues and eigenvectors · 1 mark · Multiple choice

If eigenvalues of a $2\times2$ matrix P are 4 and 2, the eigenvalues of $P^{-1}$ are

  1. 0, 0
  2. 0.0625, 0.25
  3. 0.25, 0.5
  4. 2, 4

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Correct answer: (C) 0.25, 0.5

Explanation

If $\lambda$ is an eigenvalue of an invertible matrix $P$ with eigenvector $\mathbf v$, then
$$P\mathbf v=\lambda\mathbf v\;\Rightarrow\;\mathbf v=\lambda P^{-1}\mathbf v\;\Rightarrow\;P^{-1}\mathbf v=\frac1\lambda\mathbf v .$$

So $P^{-1}$ has the same eigenvectors with **reciprocal eigenvalues**:
$$\frac14=0.25,\qquad\frac12=0.5 .$$

Answer **0.25, 0.5** (C).