8. Sum of the eigenvalues of the matrix
2 2 −1
5 3 1
1 4 −2
is ________. (1m)
(a) 1
(b) 2
(c) 3
(d) None of the above
Sum of Eigenvalues of a 3×3 Matrix Using Trace Property
Correct Answer: (c) 3
Step-by-Step Solution
Given Matrix:
\[
A =
\begin{bmatrix}
2 & 2 & -1 \\
5 & 3 & 1 \\
1 & 4 & -2
\end{bmatrix}
\]
For any square matrix, the sum of its eigenvalues (counting multiplicities) equals the trace of the matrix,
i.e., the sum of its diagonal entries.
Trace of A:
tr(A) = 2 + 3 + (−2) = 3
Therefore, the sum of the eigenvalues of A is 3.
Option Analysis
- (a) 1: Incorrect — does not match the trace.
- (b) 2: Incorrect — does not match the trace.
- (c) 3: Correct — equals the trace of the matrix.
- (d) None of the above: Incorrect — because option (c) is correct.
Introduction
Finding the sum of eigenvalues of a matrix is a common question in linear algebra and competitive exams.
Knowing the trace property allows you to answer such questions instantly,
without solving the full characteristic equation.
This shortcut is especially useful for 3×3 matrices where explicit computation of eigenvalues can be lengthy.
Concept: Sum of Eigenvalues and Trace
For an n × n square matrix A with eigenvalues λ₁, λ₂, …, λₙ:
λ₁ + λ₂ + ⋯ + λₙ = tr(A)
Here, tr(A) denotes the trace of matrix A — the sum of its diagonal elements.
This relation holds even if the matrix is not diagonalizable, making it a robust and universally valid property.
Applying the Property to the Given Matrix
For
\[
A =
\begin{bmatrix}
2 & 2 & -1 \\
5 & 3 & 1 \\
1 & 4 & -2
\end{bmatrix}
\]
The diagonal entries are 2, 3, and −2.
Compute the trace:
tr(A) = 2 + 3 − 2 = 3
Hence, the sum of the eigenvalues of A is 3, by the trace–eigenvalue property.
Option-Wise Explanation
- (a) 1: Would imply the trace is 1, but the actual trace is 3. False.
- (b) 2: Inconsistent with the trace; no valid eigenvalue set for this matrix sums to 2.
- (c) 3: Matches the trace exactly — correct sum of eigenvalues.
- (d) None of the above: Rejected, since the correct value (3) appears in option (c).
Final Answer
Using the trace property provides a fast and exam-friendly approach to eigenvalue problems.
The final answer is:
Sum of eigenvalues = tr(A) = 3
