A square matrix A is called nilpotent if Ak 0 for some posi

A square matrix A is called nilpotent if A^k = 0 for some positive integer k. The smallest such k is called the index of nil potency. Use a calculator, or by hand, compute -A^2, A^3, etc. to determine the index of nil potency of [0 1 1 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0].

Solution

Here I show that A^5 = 0. i..e k = 5.

The powers of A have been determined using a computer program in matlab and they are given as follows:

A = [0 1 1 1 1; 0 0 1 1 1; 0 0 0 1 1; 0 0 0 0 1; 0 0 0 0 0]

A =

0 1 1 1 1
0 0 1 1 1
0 0 0 1 1
0 0 0 0 1
0 0 0 0 0

>> A^2

ans =

0 0 1 2 3
0 0 0 1 2
0 0 0 0 1
0 0 0 0 0
0 0 0 0 0

>> A^3

ans =

0 0 0 1 3
0 0 0 0 1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0

>> A^4

ans =

0 0 0 0 1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0

>> A^5

ans =

0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0