39. What are the eigenvalues of the following matrix?
[ 1 1
−2 4 ]
(A) 2 and 3
(B) −2 and 3
(C) 2 and −3
(D) −2 and −3
Introduction
Finding eigenvalues of a matrix is a core topic in linear algebra and is
frequently tested in exams such as JEE, JAM, and university-level mathematics.
In this article, we find the eigenvalues of the matrix
[1 1; −2 4] using the characteristic equation method.
Given Matrix
A =
| 1 1 |
| −2 4 |
Step 1: Characteristic Equation
Eigenvalues are obtained by solving:
det(A − λI) = 0
Where I is the 2×2 identity matrix.
Step 2: Form A − λI
A − λI =
| 1 − λ 1 |
| −2 4 − λ |
Step 3: Compute the Determinant
det(A − λI) = (1 − λ)(4 − λ) − (1)(−2)
= λ2 − 5λ + 6
Step 4: Solve the Equation
λ2 − 5λ + 6 = 0
(λ − 2)(λ − 3) = 0
Hence,
λ = 2, 3
Final Answer
The eigenvalues of the matrix are 2 and 3.
Option Analysis
Option (A): 2 and 3 ✅
Correct. These are the roots of the characteristic equation.
Option (B): −2 and 3 ❌
The trace of the matrix is 1 + 4 = 5, but −2 + 3 = 1. Hence incorrect.
Option (C): 2 and −3 ❌
The determinant of the matrix is 6, but 2 × (−3) = −6. Hence incorrect.
Option (D): −2 and −3 ❌
Both trace and determinant conditions fail for this option.
Verification Using Properties
Trace: 1 + 4 = 5 = 2 + 3
Determinant: 1 × 4 − 1 × (−2) = 6 = 2 × 3
Hence, the eigenvalues are verified.
Why Other Options Fail
Options (B), (C), and (D) do not satisfy the fundamental properties of eigenvalues,
namely that their sum equals the trace and their product equals the determinant.
Conclusion
The eigenvalues of matrix [1 1; −2 4] are
2 and 3. This result is obtained using the characteristic equation
and verified using trace and determinant properties, confirming
Option (A) as the correct answer.
