Allied & ICAR Exam Bank • ALLIED_1043NAT • 2 Marks
Section 1: Engineering Mathematics
Linear Algebra
Subtopic: Trace and determinant
Question Statement:
A $3 \times 3$ matrix $A$ has $\text{Trace}(A) = 6$ and $\det(A) = 6$. If one eigenvalue is $1$, the other two eigenvalues are $\lambda_2$ and $\lambda_3$. The value of $\lambda_2^2 + \lambda_3^2$ is:
Answer Key:
13
Step-by-Step Solution & Explanation
Sum of eigenvalues $= 1 + \lambda_2 + \lambda_3 = 6 \implies \lambda_2 + \lambda_3 = 5$. Product of eigenvalues $= 1 \times \lambda_2 \times \lambda_3 = 6 \implies \lambda_2 \lambda_3 = 6$. The roots are $2$ and $3$. Thus $\lambda_2^2 + \lambda_3^2 = 2^2 + 3^2 = 4 + 9 = 13$.