Operator Example Needed

[cubixguy77]

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.

Give an example of an operator whose matrix with respect to some basis contains only nonzero numbers on the diagonal, but the operator is not invertible.

[HallsofIvy]

Did you spend a lot of time on this? The "operator" T(x,y)= (y, x) has matrix
[0 1]
[1 0]
but is obviously invertible. (in fact its inverse is itself.)

The operator T(x,y)= (x+y,y+x) has matrix
[1 1]
[1 1]
but is obviously not invertible (because T(1,2)= T(2,1)).