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## Homework Statement

Prove that

(a b

c d)

is a unit in the ring M(R) if and only if ad-bc !=0. In this case, verify that its inverse is

(d/t -b/t

-c/t a/t)

where t= ad-bc.

## Homework Equations

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

(e f

g h)

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.

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