Well, if you take
what does that tell you?
if , what are the diagonal elements of the two matrices on the left, and the matrix on the right?
the equality sign says they are the same matrices, and matrix equality means every entry is equal.
So far as I know, the definition of a matrix inverse that is typically used is the one I gave in Post # 6. The inverse of 5 is that number such that, when you multiply it by 5, you get 1. So that'd be 1/5, because 5(1/5)=(1/5)5=1. Similarly, the inverse of
is
because
I'm not sure I understand your definition of the matrix inverse. What does the asterisk mean in your definition?