# Minimal and characteristic polynomial

1. Feb 15, 2005

### cateater2000

Find the characteristic and minimal polynomials of
A=[[0,1,1][1,0,1][1,1,0]] (3x3 matrix)

So when I work out my characteristic polynomial I went
det(xI-A)= det[[x,1,1][1,x,1][1,1,x]]
= x(x^2-1)-1(x-1)+1(1-x)
= x^3-3x+2
= (x+2)(x-1)^2
It's odd because I worked this out several times, and by Cayley Hamilton's theorem it says that a characterstic polynomial of a matrix is also an annihilating polynomial for that matrix, and I tried plugging in A to the characteristic polynomial and it didn't give me the 0 matrix.

My prof's answer for the characteristic polynomial is (t-2)(t+1)^2
and her minimal polynomail is (t+1)(t-2)

Which works.

I'm really confused, can someone please tell me what I did wrong.

2. Feb 16, 2005

### AKG

Note that it's det(xI - A), not det(xI + A), i.e. this line is wrong:

det(xI-A)= det[[x,1,1][1,x,1][1,1,x]]

3. Feb 16, 2005

### cateater2000

How is this line wrong ??

A=[[0,1,1][1,0,1][1,1,0]]
xI=[[x,0,0][0,x,0][0,0,x]]

so xI-A=[[x-0,1,1][1,x-0,1][1,1,x-0]]
=[[x,1,1][1,x,1][1,1,x]]

I'm pretty sure this looks ok

Thanks for any help in advance

Last edited: Feb 16, 2005
4. Feb 16, 2005

### HallsofIvy

Then look again!!!

xI- A=[x-0,0-1,0-1][0-1,x-0,0-1][0-1,0-1,x-0]
=[x, -1, -1][-1, x, -1][-1, -1, x].

5. Feb 16, 2005