# Become Great at Mathematics for GIS Professionals

In this note we explore matrix arithmetic for its own sake.

For a shortcut notation instead of writing a matrix as , we will write or just if the size of is understood. We understand that and .

When given a set of objects in mathematics, there are two basic questions one should ask: When are two objects equal? and How can we combine two objects to produce a third object? For the first question we have the following definition.

## Definition 1

We say that two matrices and are equal, written , provided they have the same size and their corresponding entries are equal, that is, their sizes are both and for each and , .

## Example 1

- Let and . Since the size of is

and that of is , . Do note, however, that the entries of and are the same. - Find all values of and so that . We see that the size of each matrix is . So we set the corresponding entries equal:
We see that and . From , we get that must be . From , we get that and so . Thus is also .

As for the second question, we have been doing this for quite a while now: Adding, subtracting, multiplying, and dividing(when possible) real numbers. So how can we add and subtract two matrices? Eventually we will multiply matrices, but for now we consider another multiplication. Here are the definitions.