###### Example 4.2.2

If a matrix has the characteristic equation

\begin{equation*}
(4-\lambda)^2(-5-\lambda)(1-\lambda)^7(3-\lambda)^2 = 0\text{,}
\end{equation*}

then that matrix has four eigenvalues: \(\lambda=4\) having multiplicity 2; \(\lambda=-5\) having multiplicity 1; \(\lambda=1\) having multiplicty 7; and \(\lambda=3\) having multiplicty 2. The degree of the characteristic polynomial is the sum of the multiplicities \(2+1+7+2 = 12\) so this matrix must be a \(12\times12\) matrix.