A real invertible 33 matrix M has eigenvalues i and eigenvectors e i, i=1,2,3. Which statement is correct?

$M|e_i\rangle=\lambda_i^{-1}|e_i\rangle$

$M^{-1}|e_i\rangle=\lambda_i|e_i\rangle$

The eigenvalues of M and its inverse are unrelated.

GATE 2020 PH – Question 12

Mathematical Physics · Matrices: similarity transformations, diagonalization, eigenvalues and eigen vectors · 1 mark · Multiple choice

A real invertible $3\times3$ matrix M has eigenvalues $\lambda_i$ and eigenvectors $|e_i\rangle$, $i=1,2,3$. Which statement is correct?

$M|e_i\rangle=\lambda_i^{-1}|e_i\rangle$
$M^{-1}|e_i\rangle=\lambda_i^{-1}|e_i\rangle$
$M^{-1}|e_i\rangle=\lambda_i|e_i\rangle$
The eigenvalues of M and its inverse are unrelated.

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Show answer and explanation

Correct answer: (B) $M^{-1}|e_i\rangle=\lambda_i^{-1}|e_i\rangle$

Explanation

From $M|e_i\rangle=\lambda_i|e_i\rangle$, multiply by $M^{-1}$ to obtain $|e_i\rangle=\lambda_iM^{-1}|e_i\rangle$. Invertibility ensures $\lambda_i\ne0$, so $M^{-1}|e_i\rangle=\lambda_i^{-1}|e_i\rangle$. The eigenvectors are shared and eigenvalues reciprocate. Select B.

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