Question

For the matrix \[ \begin{bmatrix} 2 & 1 & 1 \\ 0 & 2 & 1 \\ 1 & 0 & 1 \end{bmatrix}, \] an Eigen vector among the following vectors is

• A. \(\begin{bmatrix} 1
  -2 \\ 2 \end{bmatrix}\)
• B. \(\begin{bmatrix} 5
  -2 \\ -1 \end{bmatrix}\)
• C. \(\begin{bmatrix} 8
  4 \\ 4 \end{bmatrix}\)
• D. \(\begin{bmatrix} 0 \\ 5 \\ 5 \end{bmatrix}\)

Hint:

To verify an eigenvector, check if multiplying the matrix yields a scalar multiple of the vector.

Answer 1

The Correct Option is C

To verify if a vector \(v\) is an eigenvector of a matrix \(A\), we check if \(A v = \lambda v\) for some scalar \(\lambda\).
Let \[ A = \begin{bmatrix} 2 & 1 & 1 \\ 0 & 2 & 1 \\ 1 & 0 & 1 \end{bmatrix}, \quad v = \begin{bmatrix} 8 \\ 4 \\ 4 \end{bmatrix} \] Then, \[ Av = \begin{bmatrix} 2\cdot8 + 1\cdot4 + 1\cdot4 \\ 0\cdot8 + 2\cdot4 + 1\cdot4 \\ 1\cdot8 + 0\cdot4 + 1\cdot4 \end{bmatrix} = \begin{bmatrix} 20 \\ 12 \\ 12 \end{bmatrix} = 2.5 \cdot \begin{bmatrix} 8 \\ 4 \\ 4 \end{bmatrix} \Rightarrow \text{Eigenvalue } \lambda = 2.5 \]