Question

The set of eigenvalues of which one of the following matrices is NOT equal to the set of eigenvalues of

\[ \begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix} \]

• A. \( \begin{pmatrix} 1 & 4 \\ 3 & 2 \end{pmatrix} \)
• B. \( \begin{pmatrix} 3 & 4 \\ 2 & 1 \end{pmatrix} \)
• C. \( \begin{pmatrix} 3 & 4 \\ 2 & 2 \end{pmatrix} \)
• D. \( \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \)

Hint:

To find the eigenvalues of a matrix, solve the characteristic equation \( \det(A - \lambda I) = 0 \).

Answer 1

The Correct Option is D

Step 1: Find the eigenvalues of the matrix \( A = \begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix} \).
The characteristic equation is \( \det(A - \lambda I) = 0 \), where \( I \) is the identity matrix and \( \lambda \) is the eigenvalue. The matrix \( A - \lambda I \) is:

\[ A - \lambda I = \begin{pmatrix} 1 - \lambda & 4 \\ 2 & 3 - \lambda \end{pmatrix}. \]

The determinant is:

\[ (1 - \lambda)(3 - \lambda) - 8 = \lambda^2 - 4\lambda - 5 = 0. \]

Solving for \( \lambda \), we get the eigenvalues \( \lambda = 5 \) and \( \lambda = -1 \).
Step 2: Compute the eigenvalues of the given matrices.
- For \( \begin{pmatrix} 1 & 4 \\ 3 & 2 \end{pmatrix} \), the eigenvalues are \( \lambda = 5 \) and \( \lambda = -1 \).
- For \( \begin{pmatrix} 3 & 4 \\ 2 & 1 \end{pmatrix} \), the eigenvalues are \( \lambda = 5 \) and \( \lambda = -1 \).
- For \( \begin{pmatrix} 3 & 4 \\ 2 & 2 \end{pmatrix} \), the eigenvalues are \( \lambda = 4 \) and \( \lambda = 1 \), which do not match.
- For \( \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \), the eigenvalues are \( \lambda = 5 \) and \( \lambda = -1 \).
Step 3: Conclusion.
The matrix \( \begin{pmatrix} 3 & 4 \\ 2 & 2 \end{pmatrix} \) does not have the same eigenvalues as \( \begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix} \). Thus, the correct answer is \( \boxed{(C)} \).