[TBag]

But that's not necessarily true. If you have a 2 x 2 matrix, the possible matrices I see working in addition to the identity matrix are

Err, have to edit these made a mistake

-1 0
0 -1

0 1
1 0

0 -1
-1 0

For example, if your matrix is [0 -1 : -1 0], multiplying by it's transpose is

0(0) + (-1)(-1) (0)(-1) + (-1)(0)
0(-1) + (0)(-1) (-1)(-1) + (0)(0)

Which is [1 0 : 0 1], an identity 2 x 2 matrix. I dunno if I"m doing something wrong, but I don't think we can prove it's definately an identity matrix