![Q.11 The eigenvalues of A = [ 1 -4 2 -3 ] are (A) 2±i (B) -1, -2 (C) -1±2i (D) non-existent](https://www.letstalkacademy.com/wp-content/uploads/2026/01/slide_11-1.jpg)
Q.11 The eigenvalues of A =
[ 1 -4
2 -3 ]
are
(A) 2±i
(B) -1, -2
(C) -1±2i
(D) non-existent
Eigenvalues of a 2×2 Matrix – Step-by-Step Solution with Explanation
Finding the eigenvalues of a matrix is a core concept in linear algebra and is
frequently tested in competitive exams such as GATE, IIT-JAM, NET, and engineering entrance tests.
This article explains how to compute eigenvalues of a given 2×2 matrix and analyze all answer options.
Given Matrix
A =
[ 1 -4 ]
[ 2 -3 ]
Step 1: Eigenvalue Formula
Eigenvalues λ of a matrix A are obtained from the
characteristic equation:
det(A − λI) = 0
where I is the identity matrix.
Step 2: Form the Characteristic Matrix
A − λI =
[ 1 − λ -4 ]
[ 2 -3 − λ ]
Step 3: Calculate the Determinant
det(A − λI) = (1 − λ)(−3 − λ) − (−4)(2)
= (1 − λ)(−3 − λ) + 8
Expanding:
(1 − λ)(−3 − λ) = λ2 + 2λ − 3
λ2 + 2λ − 3 + 8 = λ2 + 2λ + 5
Step 4: Solve the Characteristic Equation
λ2 + 2λ + 5 = 0
Using the quadratic formula:
λ = [ −2 ± √(4 − 20) ] / 2
λ = [ −2 ± √(−16) ] / 2
λ = −1 ± 2i
Correct Answer
Option (C): −1 ± 2i
Final Conclusion
By solving the characteristic equation of the given matrix, we conclude that the eigenvalues are
complex conjugates.
Final Answer: −1 ± 2i (Option C)
