- #1
RossH
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Homework Statement
Prove that the only invertible nxn idempotent matrix is the identity matrix.
Homework 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
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!