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. However, in the first
part of this book, the field will typically be either the real numbers or the complex
A matrix is a rectangular array of numbers. Several of them are referred to as matrices. For
example, here is a matrix.
This matrix is a 3 × 4 matrix because there are three rows and four columns. The first row
the second row is
and so forth. The first column is
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 Row
atholic. The symbol,
refers to a matrix in which the
denotes the row and the j
denotes the column. Using this notation on the above matrix, a23
There are various operations which are done on matrices. They can sometimes be added,
multiplied by a scalar and sometimes multiplied. To illustrate scalar multiplication, consider the
The new matrix is obtained by multiplying every entry of the original matrix by the given scalar.
If A is an m × n matrix −A is defined to equal
Two matrices which are the same size can be added. When this is done, the result is the
matrix which is obtained by adding corresponding entries. Thus
Two matrices are equal exactly when they are the same size and the corresponding entries are
because they are different sizes. As noted above, you write
for the matrix
entry is cij.
In doing arithmetic with matrices you must define what happens in
terms of the cij
sometimes called the entries of the matrix or the components of the
The above discussion stated for general matrices is given in the following definition.
Definition 2.1.1 Let A =
be two m × n matrices. Then A
for cij = aij + bij. Also if x is a scalar,
where cij = xaij. The number Aij will typically refer to the ijth 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 cij = aij + bij is just saying
that you add corresponding entries to get the result of summing two matrices as discussed
Note that there are 2 × 3 zero matrices, 3 × 4 zero matrices, etc. In fact for every size there is a
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
the commutative law of addition,
the associative law for addition,
the existence of an additive identity,
the existence of an additive inverse. Also, for α,β scalars, the following also hold.
The above properties, 2.1 - 2.8 are known as 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 2.1.2 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
referred to as a row vector.
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.
First consider the problem of multiplying an m × n matrix by an n × 1 column vector.
Consider the following example
Thus it is what is called a linear combination of the columns. These will be discussed more
later. Motivated by this example, here is the definition of how to multiply an m×n matrix by an
n × 1 matrix (vector).
Definition 2.1.3 Let A = Aij be an m × n matrix and let v be an n × 1 matrix,
where ai is an m × 1 vector. Then Av, written as
is the m × 1 column vector which equals the following linear combination of the columns.
If the jth column of A is
then 2.9 takes the form
Thus the ith entry of Av is ∑
j=1nAijvj. Note that multiplication by an m × n matrix takes an
n × 1 matrix, and produces an m × 1 matrix (vector).
Here is another example.
First of all, this is of the form
and so the result should be a
the inside numbers cancel. To get the entry in the second row and first and only column, compute
You should do the rest of the problem and verify
With this done, the next task is to multiply an m × n matrix times an n × p matrix. Before
doing so, the following may be helpful.
Definition 2.1.5 Let A be an m × n matrix and let B be an n × p matrix. Then B is of the
where bk is an n × 1 matrix. Then an m × p matrix AB is defined as follows:
where Abk is an m × 1 matrix. Hence AB as just defined is an m × p matrix. For
Example 2.1.6 Multiply the following.
The first thing you need to check before doing anything else is whether it is possible to do the
multiplication. The first matrix is a 2 × 3 and the second matrix is a 3 × 3. Therefore, is it possible
to multiply these matrices. According to the above discussion it should be a 2 × 3 matrix of the
You know how to multiply a matrix times a vector and so you do so to obtain each of the three
Here is another example.
Example 2.1.7 Multiply the following.
First check if it is possible. This is of the form
The inside numbers do not
match and so you can’t do this multiplication. This means that anything you write will be
absolute nonsense because it is impossible to multiply these matrices in this order. Aren’t they the
same two matrices considered in the previous example? Yes they are. It is just that here they are
in a different order. This shows something you must always
remember about matrix
Matrix multiplication is not commutative. This is very different than multiplication of