Find the determinant

I am trying to find the eigenvalues of a matrix and I cannot remember how to find the determinant of $A-\lambda I$:

$$\pmatrix{1-\lambda& 2& 1 \\ 2& -\lambda & -2 \\ -1& 2& 3-\lambda}$$

I know the rules for finding the determinant of an $m \times n$ matrix.

1. $\det A = -\det A'$ when switching a row
2. $\det A = \det A'$ when performing the replacement row operating
3. $\det A = \dfrac{1}{\lambda}\det A'$ 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.

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Does this help Det of $nxn$ Matrix and Examples? – Amzoti Dec 16 '12 at 2:46
For a $4 \times 4$ or larger matrix, unless there is special structure that makes it easy, finding the determinant by hand is really a waste of time. Use a computer algebra system or Wolfram Alpha: e.g. wolframalpha.com/input/… – Robert Israel Dec 16 '12 at 3:35

My favorite method for finding $3\times3$ determinants is shown here. For larger matrices, you'll have to resort to the more general formula given in the Wikipedia article on determinants.
He wanted a general approach, not just $3\times3$. – Amzoti Dec 16 '12 at 2:50