Find the eigenvalues of the matrix [[3, 1], [1, 3]]

Published on December 1, 2024

Linear-Algebra

The eigenvalues of the matrix [[3, 1], [1, 3]] are 4 and 2.

Question

eigenvalues of [[3, 1], [1, 3]]

Answer

Here’s how to find the eigenvalues of the matrix

Find the characteristic equation. The characteristic equation is given by , where is the matrix, represents the eigenvalues, and is the identity matrix.

In our case, and . So, .
Calculate the determinant. The determinant of a 2x2 matrix is . Therefore, the determinant of is:

.
Solve the characteristic equation. We set the determinant equal to zero and solve for :

This is a quadratic equation. We can factor it as:
Find the eigenvalues. The solutions to the equation are and . These are the eigenvalues of the matrix.

Therefore, the eigenvalues of the matrix

are 4 and 2.

Final Answer: The final answer is

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