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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.

  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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    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. http://www.wolframalpha.com/input/?i=characteristic+polynomial++[[+1%2C+2%2C+1]%2C+[2%2C+0%2C+-2]%2C+[-1%2C+2%2C+3+]]2012-12-16

2 Answers 2

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The standard way to calculate the determinant of small matrices manually is the Laplace expansion: http://en.wikipedia.org/wiki/Laplace_expansion#Examples

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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.

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    He wanted a general approach, not just $3\times3$.2012-12-16