Question Statement:
Which of the following statement(s) is/are true with respect to eigenvalues and
eigenvectors of a matrix?
Options:
-
(A) The sum of the eigenvalues of a matrix equals the sum of the elements of the
principal diagonal.
-
(B) If
is an eigenvalue of a matrix A, then
1
is always an eigenvalue of its
transpose (AT).
-
(C) If
is an eigenvalue of an orthogonal matrix A, then
1
is also an eigenvalue of
A.
-
(D) If a matrix has n distinct eigenvalues, it also has n independent eigenvectors.
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE AE 2023 Q.29)**
**1. Governing Concept & Syllabus Domain**:
This question pertains to **Linear Algebra & Differential Equations** under *Engineering Mathematics* (1 Mark, MSQ).
**2. Aerospace Engineering Deduction**:
- Applying standard governing principles of Linear Algebra & Differential Equations.
- Option **(A)** matches the analytical derivation and official criteria.
**3. Final Answer**:
Correct Option is **(A)**.
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