Let be the category for matrices such that , and . We can define a composition function using knowledge of matrix multiplication:
Now let , , and in that . From the composition of matrices, we can now imply that the category is associative:
Here is the communative diagram for the associativity:
Therefore we may show that the identity where and :
We have defined a category of matrices. (hopefully…)
Note: Please let me know of any errors I have made as I want to be able to correct them as I learn the concepts.