Exercise 934Linear algebra › Eigenvalues and diagonalisationL1Added 3 Nov 2026

A 3×3 matrix with distinct eigenvalues

#eigenvalues#eigenvectors#3x3 matrices

Statement

Find the eigenvalues of
$A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}, \qquad B = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 3 \end{pmatrix}.$

Hint

Both matrices are triangular.

Show solutionHide solution

Solution

Matrix $A$ . $A$ is diagonal, so its eigenvalues are its diagonal entries: $1,2,3$ .

Matrix $B$ . $B$ is upper triangular with diagonal entries $1,2,3$ , so its eigenvalues are also $1,2,3$ .

Remark. The eigenvalues of a triangular matrix are its diagonal entries. The two matrices $A$ and $B$ have the same eigenvalues but are not similar (the first is diagonalizable, the second is not, as we will see).

A 3×3 matrix with distinct eigenvalues

Statement

Solution

More problems in Eigenvalues and diagonalisation