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The characteristic equation of a matrix A is |A − I.x| = 0, where I is the identity matrix.
So, for example, if A = (

1	2
3	4

) then the characteristic equation is

        1 2 3 4     −     1 0 0 1     x     = 0

        1 2 3 4     −     x 0 0 x         = 0

    (1− x ) 2 3 (4− x )     = 0

(1− x)(4− x) − 2.3 = 0

4 −5 x + x2 −6 = 0

x2 −5 x −2 = 0.

The solutions to the characteristic equation are eigenvalues of the matrix.