is a unit in the ring M(R) if and only if ad-bc !=0. In this case, verify that its inverse is
where t= ad-bc.
An element a in a ring R with identity is called a unit if there exists u in R
s.t. au = 1 = ua (where 1 is the identity of the ring). In thise case
the element u is called the multiplicative inverse of a and is denoted a^-1.
The Attempt at a Solution
I took an arbitrary element from M(R) 2x2, say
and I know that if I multiply my unit times that, then I should get
the identity matrix.
(a b x (e f ) = (1 0
c d) . g h ) . 0 1)
which gives me four equations
ae + bg = 1 ; ce + dg = 0
af + bh = 0 ; cf + dh = 1
This is where I became lost. I started substituting things, and it became
a mess. I guess this question is really about showing where the
determinant comes from. Please help, thank you.