Find a nonzero 3 times 3 matrix whose eigenvalues are all ze

Find a nonzero 3 times 3 matrix whose eigenvalues are all zero. Is this matrix diagonalizable? Why or why not?

Solution

We know that a nilpotent matrix has 0 eigenvalues. Thus, an example of a non-zero 3x3 matrix with all its eigenvalues 0 is the nilpotent 3x3 matrix A =

5

-3

2

15

-9

6

10

-6

9

This matrix is not diagonalizable as it does not have 3 distinct eigenvalues.

5	-3	2
15	-9	6
10	-6	9