Invertible Operator with 0's on the Diagonal: What Conditions are Needed?

[cubixguy77]

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.

The Attempt at a Solution

I think the operator will not have an upper-triangular matrix since it would then not be invertible.

 

[Dick]

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

 

[cubixguy77]

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]

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.

 

[cubixguy77]

our class was taught such that we're leaving determinants out until the end of the year...
so that doesn't 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) ?

 

[PingPong]

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]

our class was taught such that we're leaving determinants out until the end of the year...
so that doesn't 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) ?

You must know SOME way to find the inverse of a matrix. Take M=[[0,a],[b,0]], put it into that method and figure out what conditions a and b need to satisfy for it to have an inverse.