Question

The matrix \[ A = \begin{bmatrix} 4 & 3 \\ 9 & -2 \end{bmatrix} \] has eigenvalues -5 and 7. The eigenvector(s) is/are _________

• A. ( \begin{bmatrix} 1 \\ 1 \end{bmatrix} \)
• B. ( \begin{bmatrix} 3 \\ 4 \end{bmatrix} \)
• C. ( \begin{bmatrix} 2 \\ -6 \end{bmatrix} \)
• D. ( \begin{bmatrix} 2 \\ 8 \end{bmatrix} \)

Hint:

To find eigenvectors, solve \( (A - \lambda I)v = 0 \) for each eigenvalue \( \lambda \). The solutions give the corresponding eigenvectors.

Answer 1

The Correct Option is A, C

Step 1: Understanding the eigenvector calculation.
To find the eigenvectors corresponding to the given eigenvalues \( \lambda = -5 \) and \( \lambda = 7 \), we solve for \( v \) in the equation \( (A - \lambda I) v = 0 \), where \( I \) is the identity matrix. We calculate the eigenvectors for each eigenvalue.
Step 2: Eigenvectors corresponding to \( \lambda = -5 \).
For \( \lambda = -5 \), solving \( (A + 5I)v = 0 \) gives the eigenvector \( \begin{bmatrix} 1 \\1 \end{bmatrix} \).
Step 3: Eigenvectors corresponding to \( \lambda = 7 \).
For \( \lambda = 7 \), solving \( (A - 7I)v = 0 \) gives the eigenvector \( \begin{bmatrix} 2 \\ -6 \end{bmatrix} \).
Step 4: Conclusion.
The correct eigenvectors are \( \begin{bmatrix} 1 \\ 1 \end{bmatrix} \) and \( \begin{bmatrix} 2 \\ -6 \end{bmatrix} \). Thus, the correct answers are (A) and (C).