9867

Matrix Representation of the Addition Group

The real numbers with the operation of addition are commonly cited as an elementary example of a group. The requirements of closure, associativity, and the existence of an inverse are all fulfilled. It is well known that groups can be represented by matrices, with the group structure reflected in the corresponding matrix multiplication. So how can one reconcile representing addition by multiplication?
The answer is provided by a simple relation involving multiplication of 2×2 matrices:
.
In this Demonstration, values of and are limited to integers between and for neatness, but the result applies to all real and complex numbers.

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DETAILS

Remember the rule for multiplication of matrices: .
Snapshot 1: two ways of representing
Snapshot 2: every element (number or matrix) has an inverse
Snapshot 3: the identity element—the number 0 or the matrix
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