(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that the only invertible nxn idempotent matrix is the identity matrix.

2. Relevant equations

idempotent: given an nxn matrix, A^2=A

invertible: There exists a matrix B such that AB=BA=I

I, or identity matrix: for all matrices A, AI=IA=A

3. The attempt at a solution

So far I haven't gotten very far:

Suppose A is an nxn matrix.

For A to be invertible there exists a matrix B such that AB=BA=I.

For A to be idempotent A^2=A.

Otherwise, I've kind of mulled it over and gotten this far:

To be

invertible it has to have a determinant != 0. So, I think that I have

to show that the determinant of everything else that is indempotent is

0, which would mean widely defining the criteria for a matrix to be

indempotent and showing that the two are mutually exclusive. The

problem is I'm not quite sure about the properties of indempotent

matrices and, since we don't have a formula for the determinant of a

given nxn matrix, I'm not quite sure what to do.

So far, I have not covered and am not allowed to use rank or eigenvalues in my class.

Thanks!

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# Linear algebra:Prove that the only invertible nxn idempotent matrix is I

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