Q.12 If the eigenvalues of a 2×2 matrix P are 4 and 2, then the eigenvalues of the
matrix P^-1 are
(A) 0, 0
(B) 0.0625, 0.25
(C) 0.25, 0.5
(D) 2, 4
Why Reciprocals?
If Pv = λv for eigenvalue λ and eigenvector v, multiply both sides by P⁻¹:
P⁻¹(Pv) = λP⁻¹v
This simplifies to v = λP⁻¹v, or:
P⁻¹v = (1/λ)v
Thus eigenvalues of P⁻¹ are reciprocals of eigenvalues of P.
Given λ = 4, 2 → reciprocals are:
- 1/4 = 0.25
- 1/2 = 0.5
This holds because P is invertible (nonzero eigenvalues).
Options Analysis
| Option | Eigenvalues | Explanation |
|---|---|---|
| (A) | 0, 0 | Incorrect. Would imply P is singular, but eigenvalues are nonzero. |
| (B) | 0.0625, 0.25 | Incorrect. Reciprocals are 1/4 and 1/2, not 1/16 and 1/4. |
| (C) | 0.25, 0.5 | Correct. Direct reciprocals of 4 and 2. |
| (D) | 2, 4 | Incorrect. Same as P; inverse requires reciprocals. |
Exam Relevance
This concept applies to any invertible square matrix and is frequently tested in:
- JEE Main / JEE Advanced
- GATE Engineering Mathematics
- Linear Algebra courses (Eigenvalues, Diagonalization, Cayley-Hamilton)
