According to Wikipedia-
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
Now, when we work with matrices we define addition,subtraction,multiplication operations (not division) on matrices.
So,going by the above definition, does a matrix qualify to be a field as we define the mentioned operations on them?
I have two doubts-
- The definition mentions "...a field is a set..." . I know a set is any well defined collection of numbers. A matrix is also a collection of numbers but not well defined. Is this a reason why we cannot call a matrix a set?
- Division of matrices is not defined. Is this also a reason that eliminates the possibilities of a matrix to be called a set?
Thanks for any help!