Question

Let \(A = \begin{pmatrix} -1 & 3 & 5 \\ -3 & k & 6 \\ 0 & 0 & 3 \end{pmatrix}\). If the eigenvalues of \(A\) are \(3, -1 + 3i, -1 - 3i\), where \(i = \sqrt{-1}\), then the value of \(k\) is _______

• A. \(-1\)
• B. \(-3\)
• C. \(0\)
• D. \(3\)

Hint:

Using the Trace-Eigenvalue relationship (\(\text{Trace} = \sum \lambda\)) is the fastest way to solve for unknown elements in a matrix when the eigenvalues are given.
This avoids having to compute the complex characteristic equation \(\det(A - \lambda I) = 0\), saving significant time.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks us to find the value of the unknown parameter \(k\) in a given \(3 \times 3\) matrix \(A\), using the known eigenvalues of the matrix.

Step 2: Key Formula or Approach:
A fundamental theorem in linear algebra states that the trace of a square matrix (the sum of the elements on its main diagonal) is always equal to the sum of its eigenvalues:
\[ \text{Trace}(A) = \sum_{i=1}^{n} a_{ii} = \sum_{i=1}^{n} \lambda_i \] where:
\(a_{ii}\) are the diagonal elements of the matrix.
\(\lambda_i\) are the eigenvalues of the matrix.

Step 3: Detailed Explanation:
Let us compute both the trace of the matrix and the sum of the eigenvalues for the given problem:
1. Find the trace of matrix \(A\):
The diagonal elements of \(A = \begin{pmatrix} -1 & 3 & 5 \\ -3 & k & 6 \\ 0 & 0 & 3 \end{pmatrix}\) are \(a_{11} = -1\), \(a_{22} = k\), and \(a_{33} = 3\).
The trace is the sum of these elements:
\[ \text{Trace}(A) = -1 + k + 3 = k + 2 \] 2. Find the sum of the given eigenvalues:
The eigenvalues are \(\lambda_1 = 3\), \(\lambda_2 = -1 + 3i\), and \(\lambda_3 = -1 - 3i\).
Summing these eigenvalues:
\[ \sum \lambda_i = 3 + (-1 + 3i) + (-1 - 3i) \] Combine the real parts and the imaginary parts:
\[ \sum \lambda_i = 3 - 1 - 1 + 3i - 3i = 1 \] 3. Equate the trace to the sum of the eigenvalues:
\[ k + 2 = 1 \] 4. Solve for \(k\):
\[ k = 1 - 2 = -1 \] This matches Option (A) perfectly.

Step 4: Final Answer
The value of \(k\) is \(-1\), which corresponds to option (A).