Matrix field

In abstract algebra, a matrix field is a field with matrices as elements. In field theory we come across two types of fields: finite fields and infinite fields. There are several examples of matrix fields of different characteristic and cardinality.
There is a finite matrix field of cardinality "p" for each positive prime "p". One can find several finite matrix fields of characteristic "p" for any given prime number "p". In general, corresponding to each finite field there is a matrix field. Since any two finite fields of equal cardinality are isomorphic, the elements of a finite field can be represented by matrices.
Contrary to the general case for matrix multiplication, multiplication is commutative in a matrix field (if the usual operations are used). Since addition and multiplication of matrices have all needed properties for field operations except for commutativity of multiplication and existence of multiplicative inverses, one way to verify if a set of matrices is a field with the usual operations of matrix sum and multiplication is to check whether 
Examples.
1. Take the set of all formula_1 matrices of the form
with formula_3 that is, matrices filled with zeroes except for the first row, which is filled with the same real constant formula_4.
These matrices are commutative for multiplication:
The multiplicative identity is
formula_6.
The multiplicative inverse of a matrix formula_7 with formula_8 is given by formula_9 
2. The set of matrices of the form
where formula_4 and formula_12 range over the field of real numbers,
forms a matrix field which is isomorphic to the field formula_13 of complex numbers: formula_4 corresponds to the real part of the number, while formula_12 corresponds to the imaginary part. So the number formula_16, for example, would be represented as
One can easily verify that formula_18:
and also, by computing a matrix exponential, that Euler's identity, formula_20 is valid: