Question

Which of the following is/are the eigenvector(s) for the matrix given below? \[ \begin{pmatrix} -9 & -6 & -2 & -4 \\ -8 & -6 & -3 & -1 \\ 20 & 15 & 8 & 5 \\ 32 & 21 & 7 & 12 \end{pmatrix} \]

• A. \(\begin{pmatrix} -1 \\ 1 \\ 0 \\ 1 \end{pmatrix}\)
• B. \(\begin{pmatrix} 1 \\ 0 \\ -1 \\ 0 \end{pmatrix}\)
• C. \(\begin{pmatrix} -1 \\ 0 \\ 2 \\ 2 \end{pmatrix}\)
• D. \(\begin{pmatrix} 0 \\ 1 \\ -3 \\ 0 \end{pmatrix}\)

Hint:

When solving for eigenvectors, remember to substitute each eigenvalue back into the equation \( (A - \lambda I)v = 0 \) and solve the system of linear equations to find the corresponding eigenvectors.

Answer 1

The Correct Option is A, C, D

To find the eigenvectors of a matrix, we solve the characteristic equation: \[ \text{det}(A - \lambda I) = 0 \] where \( A \) is the given matrix, \( \lambda \) is the eigenvalue, and \( I \) is the identity matrix of the same dimension. For each eigenvalue \( \lambda \), we substitute it into the equation \( (A - \lambda I)v = 0 \), where \( v \) is the eigenvector corresponding to \( \lambda \). We then solve the system of equations to find the eigenvectors.

Step 1: Eigenvector for Option (A) For the matrix \( A \), the eigenvalue corresponding to the eigenvector \(\begin{pmatrix} -1 \\ 1 \\ 0 \\ 1 \end{pmatrix}\) is found to be a solution to the system. Substituting this vector into the equation results in a valid solution, making this eigenvector correct. 

Step 2: Eigenvector for Option (C) Similarly, for the vector \(\begin{pmatrix} -1 \\ 0 \\ 2 \\ 2 \end{pmatrix}\), we find that this vector satisfies the system of equations derived from the matrix and is thus another correct eigenvector. 

Step 3: Eigenvector for Option (D) The vector \(\begin{pmatrix} 0 \\ 1 \\ -3 \\ 0 \end{pmatrix}\) also satisfies the system of equations, making it another correct eigenvector for the matrix. Final Answer The correct eigenvectors for the matrix are: 
- Option (A): \(\begin{pmatrix} -1 \\ 1 \\ 0 \\ 1 \end{pmatrix}\) 
- Option (C): \(\begin{pmatrix} -1 \\ 0 \\ 2 \\ 2 \end{pmatrix}\) 
- Option (D): \(\begin{pmatrix} 0 \\ 1 \\ -3 \\ 0 \end{pmatrix}\) 

Final Answer: (A), (C), (D)