# Thread: Nilpotent Matrix

1. ## Nilpotent Matrix

Hi,

Is my working and answer to the following question correct?

Question: Show that matrix A is nilpotent and state the index of nilpotency k

$\displaystyle A = \begin {pmatrix} -2 & 1 \\ -4 & 2 \end {pmatrix}$

Definition of Nilpotent: A matrix A is said to be nilpotent if $\displaystyle A^k=0$ for some positive integer k. The smallest k is called the index of nilpotency for A

$\displaystyle A^2 = \begin {pmatrix} -2 & 1 \\ -4 & 2 \end {pmatrix}$$\displaystyle \begin {pmatrix} -2 & 1 \\ -4 & 2 \end {pmatrix}$

$\displaystyle = \begin {pmatrix} 0 & 0 \\ 0 & 0 \end {pmatrix}$

A is nilpotent, k=2

2. Looks good to me!

3. Thanks Ackbeet

4. You're welcome.