In my two-part series on equation systems, I mentioned that there is a third method using matrices. Before I use this method, I have to explain what a matrix is. There is a whole new algebra surrounding matrices (the plural of matrix), so I will only explain what is needed to solve a system of equations.
All of the algebra we have used so far is called scalar algebra. That is, it is used on scalars. A scalar is just a single number like 3, -2.7, or x. It just has a value and you cannot get any more information from it like direction. In most engineering problems, 3-dimensional space is the playground and engineers are also interested in not only the speed of an object, but also the direction it is travelling in. Scalar algebra will not suffice. Enter matrix algebra. This is sometimes equated to linear algebra, but there are differences. I will continue to use matrix algebra as that is the most appropriate at this level.
A matrix is simply an array of numbers. Here are some examples:\[
Of course, you cannot talk about a whole new field of maths without definitions. The first definition is dimension. The dimension of a matrix is its size. By convention, the dimension always indicates the number of rows first then the number of columns. So the first matrix in the examples above, is a 2 × 2 matrix because it has 2 rows and 2 columns. The second matrix is a 4 × 1 matrix. Matrices with 1 as one of the dimensions, can be called vectors as well. The third matrix is 1 × 3 and the last is 2 × 3.
It is very common to use bold typeface when using a single letter to represent a matrix so that the reader knows that they are talking about a matrix and not a scalar:\[
So I can now talk about A and you know that A is a matrix.
So there are the same types of operations for matrices as there are for scalars: adding, subtracting, multiplying, and dividing. But there are rules associated with these matrix operations that are specific to matrices.
Since I am directing the use of matrices to solving a system of equations, I will only discuss the multiplying and dividing operations as they relate to matrices. This is where I will begin in my next post.