# Cayley-Hamilton theorm

1. Jul 28, 2016

### Delta what

1. The problem statement, all variables and given/known data
I have matrix F=(11,1, -i1,2, i2,1, 12,2)
I calculated the characteristic polynomial to be x2-2x
The question is to find the inverse of F
2. Relevant equations
I am having a hard time trying to get the inverse out of this. I am used to dealing with real matrices so this may be the source of my error. I have done these with the usual case being that there was a constant term in the polynomial which one could get on the other side of the equality and get the inverse easily.

3. The attempt at a solution
I am ending up with something like X2=2x and then multiplying by x-1 and ending up with x=2. What am I missing here?

Side note (new guy question) could someone point me into the direction of an explanation of how to do math notation of this forum such as matrices and similar?

2. Jul 28, 2016

### Ray Vickson

If the characteristic polynomial is $c(x)= x^2 - 2x$ the eigenvalues are $\lambda_1 = 0$ and $\lambda_2 = 2$. Since one of the eigenvalues is zero the matrix does not have an inverse.

3. Jul 28, 2016

### Delta what

Thank you for the help!

4. Jul 28, 2016

### Delta what

Also on the side note I found that LaTeX is used to write in various notations.

5. Jul 28, 2016

### vela

Staff Emeritus
Here's a LaTeX FAQ: https://www.physicsforums.com/help/latexhelp/

The Cayley-Hamilton theorem tells you $F^2 = 2F$. When you multiplied by $F^{-1}$, you assumed the inverse exists, and this assumption leads to the conclusion that $F=2I$, which clearly contradicts what you started with. Therefore, you would conclude that the assumption was wrong. $F^{-1}$ doesn't exist.

6. Jul 28, 2016

### Delta what

Thank you for your help vela!

7. Jul 28, 2016

### Ray Vickson

8. Jul 28, 2016

### Delta what

Little bit of a different question but pertains to the same matrices. If i have a eigenvalue of zero can i still use that as a value in my diagonal matrix?

9. Jul 28, 2016

### vela

Staff Emeritus
Yes.

10. Jul 29, 2016

### epenguin

More elementary, if I understand your notation, then if you multiply the second row by i, and then add that to the first row you see that the matrix is singular.