You have now solved systems of equations by writing them in terms of an augmented matrix and then doing row operations on this augmented matrix. It turns out that such rectangular arrays of numbers are important from many other different points of view. Numbers are also called scalars. In general, scalars are just elements of some field.
A matrix is a rectangular array of numbers from a field F. For example, here is a matrix.

This matrix is a 3 × 4 matrix because there are three rows and four columns. The columns stand upright and are listed in order from left to right. The columns are horizontal and listed in order from top to bottom. The convention in dealing with matrices is to always list the rows first and then the columns. Also, you can remember the columns are like columns in a Greek temple. They stand up right while the rows just lie there like rows made by a tractor in a plowed field. Elements of the matrix are identified according to position in the matrix. For example, 8 is in position 2,3 because it is in the second row and the third column. You might remember that you always list the rows before the columns by using the phrase Rowman Catholic. The symbol,
There are various operations which are done on matrices. They can sometimes be added, multiplied by a scalar and sometimes multiplied.
Definition 4.0.1 Let A =

for c_{ij} = a_{ij} + b_{ij}. Also if x is a scalar,

where c_{ij} = xa_{ij}. The number A_{ij} will also typically refer to the ij^{th} entry of the matrix A. The zero matrix, denoted by 0 will be the matrix consisting of all zeros.
Do not be upset by the use of the subscripts, ij. The expression c_{ij} = a_{ij} + b_{ij} is just saying that you add corresponding entries to get the result of summing two matrices as discussed above.
Note that there are 2 × 3 zero matrices, 3 × 4 zero matrices, etc. In fact for every size there is a zero matrix.
With this definition, the following properties are all obvious but you should verify all of these properties are valid for A, B, and C, m × n matrices and 0 an m × n zero matrix.
 (4.1) 
the commutative law of addition,
 (4.2) 
the associative law for addition,
 (4.3) 
the existence of an additive identity,
 (4.4) 
the existence of an additive inverse. Also, for α,β scalars, the following also hold.
 (4.5) 
 (4.6) 
 (4.7) 
 (4.8) 
The above properties, 4.1  4.8 are the vector space axioms and the fact that the m×n matrices satisfy these axioms is what is meant by saying this set of matrices with addition and scalar multiplication as defined above forms a vector space.
Definition 4.0.2 Matrices which are n× 1 or 1 ×n are especially called vectors and are often denoted by a bold letter. Thus

is an n× 1 matrix also called a column vector while a 1 ×n matrix of the form
All the above is fine, but the real reason for considering matrices is that they can be multiplied. This is where things quit being banal. The following is the definition of multiplying a m×n matrix times a n× 1 vector.
Definition 4.0.3 First of all, we define the product of a 1 × n matrix and a n × 1 matrix.

If you have A an m × n matrix and B is an n × p matrix, then AB will be an m × p matrix whose ij^{th} entry is the product of the i^{th} row of A on the left with the j^{th} column of B on the right. Thus

and if B =

You can do
To see the last claim, note that the j^{th} column of AB involves b_{j} and is of the form

Here is an example.
Example 4.0.4 Compute the following product in ℤ_{5}. That is, all the numbers are interpreted as residue classes.

Doing the arithmetic in ℤ_{5}, you get
