A **nilpotent matrix (P)** is a square matrix, if there exists a positive integer ‘m’ such that P^{m} = O. In other words, matrix **P** is called nilpotent of index **m** or class **m** if P^{m} = O and P^{m-1} ≠ O. Here O is the null matrix (or zero matrix).

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## Nilpotent matrix Examples

The examples of 2 x 2 nilpotent matrices are

1.

Matrix A is a nilpotent matrix of index 2. It means that A ≠ O and A^{2} = O.

2.

Matrix B is a nilpotent matrix of index 2. It means that B ≠ O and B^{2} = O.

The examples of 3 x 3 nilpotent matrices are

1.

Matrix C is a nilpotent matrix of index 2. It means that C ≠ O and C^{2} = O.

2.

Matrix D is a nilpotent matrix of index 3. It means that D ≠ O, D^{2} = O and D^{3} ≠ O.

## Nilpotent matrix Properties

1. It is a square matrix.

2. All of its eigenvalues are zero.