Characteristic equation

The homogeneous equation has nontrivial solutions if and only if the matrix  $ (A - \lambda I)$ is not invertible. Thus, a scalar $ \lambda$ is an eigenvalue of $ A$ if and only if it satisfies the characteristic equation

$\displaystyle \det(A - \lambda I) = 0.
$

EXAMPLE 3 Construct the characteristic equation, and then find the eigenvalues for each of the following matrices.

  1. $ A = \begin{bmatrix}
2 & 3 \\
3 &-6
\end{bmatrix}$
  2. $ A = \begin{bmatrix}
5 &-2 & 6 &-1 \\
0 & 3 &-8 & 0 \\
0 & 0 & 5 & 4 \\
0 & 0 & 0 & 1
\end{bmatrix}$



Department of Mathematics
Last modified: 2005-10-21