![Q.20 One of the eigenvalues of P = [ 10 −4 18 −12 ] is 2 4 6 8](https://www.letstalkacademy.com/wp-content/uploads/2026/01/slide_20-3.jpg)
Q.20 One of the eigenvalues of
P =
[ 10 −4
18 −12 ]
is
- 2
- 4
- 6
- 8
Introduction
Eigenvalues are a fundamental topic in linear algebra and appear frequently in competitive
examinations such as GATE, IIT-JAM, CSIR-NET, and university exams. This example demonstrates
how to compute eigenvalues of a 2×2 matrix using the characteristic equation.
Given Matrix
P =
[ 10 -4
18 -12 ]
Concept Used
Eigenvalues λ of a square matrix P are obtained by solving:
det ( P − λI ) = 0
Step-by-Step Solution
Step 1: Form P − λI
P − λI =
[ 10 − λ -4
18 -12 − λ ]
Step 2: Find the Determinant
det ( P − λI ) =
(10 − λ)(−12 − λ) − (−4)(18)
Step 3: Simplify
(10 − λ)(−12 − λ) + 72
= λ2 + 2λ − 120 + 72
= λ2 + 2λ − 48
Step 4: Solve the Characteristic Equation
λ2 + 2λ − 48 = 0
(λ + 8)(λ − 6) = 0
λ = −8, 6
Final Answer
Correct Option: (C) 6
Key Takeaways
- Eigenvalues are found using det(P − λI) = 0
- 2×2 matrices lead to quadratic characteristic equations
- Always check signs carefully while solving
