GATE 2023 AE – Question 29

Correct answer: (A) The sum of eigenvalues equals the principal-diagonal sum.; (C) If $\lambda$ is an eigenvalue of an orthogonal matrix A, $1/\lambda$ is also an eigenvalue of A.; (D) A matrix with n distinct eigenvalues has n independent eigenvectors.

Explanation

Check each statement about eigenvalues and eigenvectors.

• **A. The sum of the eigenvalues equals the sum of the diagonal elements.** True: $\sum\lambda_i=\text{trace}(A)$. ✓
• **B. If $\lambda$ is an eigenvalue of $A$, then $1/\lambda$ is always an eigenvalue of $A^T$.** False. $A^T$ has the **same** eigenvalues as $A$ (equal characteristic polynomial), not their reciprocals.
• **C. For an orthogonal matrix, $1/\lambda$ is also an eigenvalue.** True. A real orthogonal matrix has eigenvalues of modulus 1, and non-real eigenvalues come in conjugate pairs $\lambda,\bar\lambda$ with $\bar\lambda=1/\lambda$; real ones are $\pm1$, which are their own reciprocals. Also $A^{-1}=A^T$ has the same eigenvalues as $A$. ✓
• **D. A matrix with $n$ distinct eigenvalues has $n$ independent eigenvectors.** True. Eigenvectors of distinct eigenvalues are linearly independent. ✓

Answer **A, C and D**.