FORM GR1268 Question 31

Question:
Of the numbers \(2,3,\) and \(5\), which are eigenvalues of the matrix \(\displaystyle\begin{pmatrix}3 & 5 & 3\\ 1 & 7 & 3\\ 1 & 2 & 8\end{pmatrix}\)?

(A) None     (B) \(2\) and \(3\) only     (C) \(2\) and \(5\) only     (D) \(3\) and \(5\) only     (E) \(2,3,\) and \(5\)

Answer:
(C)

Answer Key:
If \(\lambda\) is an eigenvalue of a square matrix \(\mathbf A\), then \(\mathbf{Av}=\lambda \mathbf v\) for some nonzero vector \(\mathbf v\).
Then \(\mathbf{Av}=\lambda \mathbf{Iv}\), so \((\mathbf A-\lambda \mathbf I)\mathbf v=\mathbf 0\).
If \(\mathbf A-\lambda \mathbf I\) is invertible, then \(\mathbf v=(\mathbf A-\lambda \mathbf I)^{-1}\mathbf 0=\mathbf 0\).
But we want \(\mathbf v\) to be a non-zero vector, so \(\mathbf A-\lambda \mathbf I\) needs to be singular.

\[\begin{align}
\begin{pmatrix}3 & 5 & 3\\ 1 & 7 & 3\\ 1 & 2 & 8\end{pmatrix}-2I&=\begin{pmatrix}1 & 5 & 3\\ 1 & 5 & 3\\ 1 & 2 & 6\end{pmatrix}\quad \text{(singular)}\\
\\
\begin{pmatrix}3 & 5 & 3\\ 1 & 7 & 3\\ 1 & 2 & 8\end{pmatrix}-3I&=\begin{pmatrix}0 & 5 & 3\\ 1 & 4 & 3\\ 1 & 2 & 5\end{pmatrix}\quad \text{(non-singular)}\\
\\
\begin{pmatrix}3 & 5 & 3\\ 1 & 7 & 3\\ 1 & 2 & 8\end{pmatrix}-5I&=\begin{pmatrix}-2 & 5 & 3\\ 1 & 2 & 3\\ 1 & 2 & 3\end{pmatrix}\quad \text{(singular)}\\
\end{align}\]