Idempotent matrix problem - HELP

[chefobg57]

idempotent matrix problem - HELP :(

Hi, I have the following problem with an idempotent matrix and I am stuck...

If A is an idempotent (A^2 = A), is E - nA invertible /E is the identity matrix and n is a real number/ and why?

I've tried with setting (E - nA)(E - nA) but it doesn't get me anywhere..
My instinct is telling me there it should be invertible in a lot of the cases, but... :/

 

[Dick]

If (E-nA) were NOT invertible, then there would be a nonzero vector v such that (E-nA)v=0. Where does that lead you?

 

[chefobg57]

I am sorry, I am quite new to Linear Algebra I am not really sure what the answer to your question should be... Can you explain a little bit more why there should be a non-zero vector such that (E-nA)v=0?

Thank you for helping me out! You're a life saver!

 

[Dick]

If M is a square matrix, then M being invertible is equivalent to the kernel or null space of M being {0}. If M is NOT invertible, then the kernel of M is not {0}. Meaning there is nonzero vector v in the kernel, hence Mv=0. You must have covered this. Can you look back in your book or notes for that kind of thing? It's pretty important.

 

[chefobg57]

I checked my notes and as far as I understand you're talking about the invertible matrix theoremy...
Can we then say that:
Assuming that E-nA is not invertible means that there will be a non-zero vector x, such that (E-nA)x=0. However, this suggests that E-nA is not row equivalent to E which then implies that E-nA is not a product of elementary matrices. This implies that E-nA is not invertible?

I'm sorry I'm such a pain but I am really lost...

Thanks again..

 

[Dick]

I checked my notes and as far as I understand you're talking about the invertible matrix theoremy...
Can we then say that:
Assuming that E-nA is not invertible means that there will be a non-zero vector x, such that (E-nA)x=0. However, this suggests that E-nA is not row equivalent to E which then implies that E-nA is not a product of elementary matrices. This implies that E-nA is not invertible?

I'm sorry I'm such a pain but I am really lost...

Thanks again..

You are making it much too hard. If (E-nA)v=0 then Ev-nAv=0. What is Ev?

 

[chefobg57]

Hm..

Well, Ev = nAv => v = nAv .. Doesn't this mean that nA=1?

 

[Dick]

Hm..

Well, Ev = nAv => v = nAv .. Doesn't this mean that nA=1?

NO! You can't divide both sides by a vector! But it does mean Av=v/n. Now operate on both sides with A again remembering A is idempotent.