I am trying to find the eigenvalues of a matrix and I cannot remember how to find the determinant of $A-\lambda I$:
\begin{equation} \pmatrix{1-\lambda& 2& 1 \\ 2& -\lambda & -2 \\ -1& 2& 3-\lambda} \end{equation}
I know the rules for finding the determinant of an $m \times n$ matrix.
- $\det A = -\det A'$ when switching a row
- $\det A = \det A'$ when performing the replacement row operating
- $\det A = \dfrac{1}{\lambda}\det A'$ when scaling a row
Then I row reduce until I have a diagonal matrix...
Do I have to use the above rules or is there a simpler way? I am not looking to use the formula for a $3\times 3$ matrix. I am looking for a more general solution so that I will be able to solve this problem for a $4 \times 4$ matrix or $5 \times 5$ matrix.