Questions

1. Let \(A = \begin{bmatrix} 3 & 0 & 1\\ 2 & 2 & 2\\ 0 & 1 & -1\end{bmatrix}\). You are given that \(\begin{bmatrix} -1 \\ 3 \\ 1 \end{bmatrix}\) is an eigenvector of \(A\) with eigenvalue \(\lambda\). What is \(\lambda\)?

   The answer is 2.

   Let \(x\) denote the vector \(\begin{bmatrix} -1 \\ 3 \\ 1 \end{bmatrix}\). Then \(Ax = \lambda x\) since we are given that $x$ is an eigenvector with eigenvalue 2.

   Now, \begin{align*} A x & = \begin{bmatrix} 3 & 0 & 1\\ 2 & 2 & 2\\ 0 & 1 & -1\end{bmatrix}\begin{bmatrix} -1 \\ 3 \\ 1 \end{bmatrix} \\ & = \begin{bmatrix} -2 \\ 6 \\ 2\end{bmatrix} \\ & = 2 \begin{bmatrix} -1 \\ 3 \\ 1\end{bmatrix} \\ & = 2 x. \end{align*} Thus, \(x\) is an eigenvector with eigenvalue 2.

2. Let \(A = \begin{bmatrix} 1 & 0 & 1\\ 0 & 2 & -2\\ 0 & 1 & 1\end{bmatrix}\). How many eigenvalues of \(A\) are complex?

   The answer is 2.

   Note that \(p_A = \left| \begin{array}{ccc} 1-\lambda & 0 & 1\\ 0 & 2 -\lambda & -2\\ 0 & 1 & 1-\lambda \end{array}\right| = (1-\lambda) ( (2-\lambda)(1-\lambda) - (-2)(1)) = (1-\lambda)(\lambda^2 - 3\lambda + 4).\)

   Hence, the eigenvalues are \(1\), \(\frac{3 + \sqrt{7}i}{2}\), and \(\frac{3 - \sqrt{7}i}{2}\), two of which are complex.