In this section, we use matrices to give a representation of complex numbers. Indeed, consider the set

We will write

Clearly, the set is not empty. For example, we have

In particular, we have

for any real numbers

**Algebraic Properties of **

**1.****Addition:**For any real numbers*a*,*b*,*c*, and*d*, we have

*M*_{a,b}+*M*_{c,d}=*M*_{a+c,b+d}.

In other words, if we add two elements of the set , we still get a matrix in . In particular, we have

-*M*_{a,b}=*M*_{-a,-b}.

**2.****Multiplication by a number:**We have

So a multiplication of an element of and a number gives a matrix in .**2.****Multiplication:**For any real numbers*a*,*b*,*c*, and*d*, we have

In other words, we have

*M*_{a,b}*M*_{c,d}=*M*_{ac-bd, ad+bc}.

This is an extraordinary formula. It is quite conceivable given the difficult form of the matrix multiplication that, a priori, the product of two elements of may not be in again. But, in this case, it turns out to be true.

The above properties infer to
a very nice structure. The next natural question to ask, in this case, is whether a nonzero element of
is invertible. Indeed, for any real numbers *a* and *b*, we have

So, if
,
the matrix *M*_{a,b} is invertible and

In other words, any nonzero element

In order to define the division in ,
we will use the inverse. Indeed, recall that

So for the set , we have

The matrix *M*_{a,-b} is called the **conjugate** of *M*_{a,b}. Note that the conjugate of the conjugate of *M*_{a,b} is *M*_{a,b} itself.

**Fundamental Equation.** For any *M*_{a,b} in ,
we have

Note that

**Remark.** If we introduce an imaginary number *i* such that *i*^{2} = -1, then the matrix *M*_{a,b} may be rewritten by

A lot can be said about ,
but we will advise you to visit the page on complex numbers.

**
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**Author**: M.A. Khamsi

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