Find Eigenvalues and Eigenvectors of 3x3 Matrix

Published on September 10, 2026

Linear-Algebra

Learn how to find the eigenvalues and eigenvectors of a 3x3 matrix by setting up the characteristic equation and solving the resulting homogeneous systems.

Question

With characteristics equation

Answer

To find the eigenvalues and eigenvectors of the matrix

, we follow these steps:

Step 1: Find the characteristic equation
The characteristic equation is given by

Expanding along the second row:

The eigenvalues are

and

(with multiplicity 2).

Step 2: Find the eigenvector for
Solve

From the second row,

. From the first row,

.
Let

, then

. The eigenvector is

Step 3: Find the eigenvectors for
Solve

Both non-zero rows simplify to

. The variable

is free.
We can choose two linearly independent vectors:

Let . Vector .
Let . Vector .

Final Answer:
Eigenvalues:

Eigenvectors:

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