Exercise

C11† Find the characteristic polynomial of the matrix A

\[ A = \begin{pmatrix} 3 & 2 & 1\\ 0 & 1 & 1 \\ 1 & 2 & 0 \\ \end{pmatrix} \]

Solution

|A - λI| = 0

\[\begin{pmatrix} 3 & 2 & 1\\ 0 & 1 & 1 \\ 1 & 2 & 0 \\ \end{pmatrix} - λ*\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}\]

=

\[\begin{pmatrix} 3 & 2 & 1\\ 0 & 1 & 1 \\ 1 & 2 & 0 \\ \end{pmatrix} - λ*\begin{pmatrix} λ & 0 & 0\\ 0 & λ & 0 \\ 0 & 0 & λ \\ \end{pmatrix}\]

=

\[\begin{pmatrix} 3-λ & 2 & 1\\ 0 & 1-λ & 1 \\ 1 & 2 & -λ \\ \end{pmatrix}\]

= 3-λ[(1-λ)(-λ) - (2)(1)] + 2[(0)(-λ) - (1)(1)] + [(0)(2) - (1)(1-λ)]

= (3-λ)(-λ+\(λ^{2}\)-2) - 2(-1) -1 + λ

= -3λ + \(3λ^{2}\) - 6 + \(λ^{2}\) - \(λ^{3}\) + 2λ +2 -1 + λ

= - \(λ^{3}\) + \(4λ^{2}\) - 5 = 0 (The characteristic polynomial)