# Nilpotent Matrix

• February 23rd 2011, 04:32 PM
sparky
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

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

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

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

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

A is nilpotent, k=2
• February 23rd 2011, 04:33 PM
Ackbeet
Looks good to me!
• February 23rd 2011, 04:38 PM
sparky
Thanks Ackbeet
• February 23rd 2011, 04:38 PM
Ackbeet
You're welcome.