Changes

Take <math>A = \begin{pmatrix}1 & 9 \\ 4 & 1\end{pmatrix}</math>

Take <math>A = \begin{pmatrix}1 & 9 \\ 4 & 1\end{pmatrix}</math>

Then <math>f_A(\lambda) = \begin{vmatrix}1-\lambda & 9 \\ 4 & 1-\lambda\end{vmatrix}</math><math>=(1-\lambda)^2-36\,</math><math>=(-5-\lambda)(7-\lambda)\,</math>.

Then <math>f_A(\lambda) = \begin{vmatrix}1-\lambda & 9 \\ 4 & 1-\lambda\end{vmatrix}</math><math>=(1-\lambda)^2-36\,</math><math>=(-5-\lambda)(7-\rho)\,</math>.

So, the roots of the characteristic polynomial are {-5, 7} - and these are the eigenvalues of the matrix. If you look at the slightly different matrix

So, the roots of the characteristic polynomial are {-5, 7} - and these are the eigenvalues of the matrix. If you look at the slightly different matrix