# Operator Example needed

1. Homework Statement
Give an example of an operator whose matrix with respect to some basis contains only 0's on the diagonal, but the operator is invertible.

3. The Attempt at a Solution
I think the operator will not have an upper-triangular matrix since it would then not be invertible.

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Dick
Homework Helper
No, what you after is not upper triangular. What are some alternatives?

i'm still unsure of what the diagonal values represent
if the diagonals are all 0, what does that tell you about the operator?

HallsofIvy
Homework Helper
No body said all the diagonals are 0, just the main diagonal. Start simple. Try a 2 by 2 matrix with 0s on the diagonal. Now, you know that a matrix is invertible as long as its determinant is not 0. Just fill in the other two numbers so the determinant is not 0.

our class was taught such that we're leaving determinants out until the end of the year...
so that doesnt tell me a whole heck of a lot
if the diagonal is all 0's, what condition would make the matrix invertible (aside from anything having to do with determinants) ?

Well if it's a two dimensional matrix, A, you could easily find the inverse of it and show that $AA^{-1}=I=A^{-1}A$. That way you wouldn't need to find its determinant.

Dick