Most of linear algebra involves mathematical objects called * matrices*. A * matrix* is a
finite rectangular array of numbers:

In this case, the numbers are elements of (or ). In general, the entries will be elements of some commutative ring or field.

I'll explain operations with matrices in the following examples. I'll discuss and prove some of the properties of these operations later on.

* Example.* (* Dimensions of
matrices*) An matrix is a matrix with m rows
and n columns. Sometimes this is expressed by saying that the * dimensions* of the matrix are .

A matrix is called an *
n-dimensional row vector*. For example, here's a 3-dimensional
row vector:

Likewise, an matrix is called an * n-dimensional column vector*. Here's a
3-dimensional column vector:

Two matrices are * equal* if they have the same
dimensions and the corresponding entries are equal. For example, if

then , , and .

* Definition.* If R is a commutative ring, then
is the set of matrices with
entries in R.

For example, is the set of matrices with real entries. is the set of matrices with entries in .

* Example.* (* Addition and
subtraction*) In this example, I'll assume that the matrices have
entries in .

You can add (or subtract) matrices by adding (or subtracting) corresponding entries.

If you are adding several matrices, you can group them any way you wish:

Note that matrix addition is * commutative*:

Symbolically, if A and B are matrices with the same dimensions, then

Of course, matrix subtraction is not commutative.

You can only add or subtract matrices with the same dimensions.

* Example.* (* Adding or
subtracting matrices over *) You add
or subtract matrices over by adding or
subtracting corresponding entries, but all the arithmetic is done in
.

For example, here are some matrix computations over :

Note that in the second example, there were some negative numbers in the middle of the computation, but the final answer was expressed entirely in terms of elements of .

* Example.* A * zero matrix*
is a matrix all of whose entries are 0. If
you add the zero matrix to another matrix A, you get A:

In symbols, if is a zero matrix and A is a matrix of the same size, then

A zero matrix is said to be an * identity
element* for matrix addition.

* Example.* You can multiply a matrix by a number
by multiplying each entry by the number. Here is an example with real
numbers:

Things work in the same way over , but all the arithmetic is done in . Here is an example over :

* Example.* To compute the *
product* of two matrices, take the dot products of
the rows of A with the columns of B. In this example, assume all the
matrices have real entries.

In order for the multiplication to work, the matrices must have compatible dimensions: The number of columns in A should equal the number of rows of B. Thus, if A is an matrix and B is an matrix, will be an matrix.

Here are two more examples, again using matrices with real entries:

Here is an example with matrices in . Remember that all the arithmetic is done in .

* Example.* If you multiply a matrix by a zero
matrix, you get a zero matrix:

In symbols, if is a zero matrix and A is a matrix compatible with it for multiplication, then

(The zero matrices here may have different sizes.)

* Example.* Write the system of linear equations

as a matrix multiplication equation.

* Example.* Write the system of equations which
correspond to the matrix equation

Multiply out the left side:

Equate corresponding entries:

* Example.* (* Identity
matrices*) There are special matrices which serve as * identities* for multiplication: The identity matrix is the square matrices with 1's down
the * main diagonal* --- the diagonal running
from northwest to southeast --- and 0's everywhere else. For example,
the identity matrix is

If I is the identity and A is a matrix which is compatible for multiplication with A, then

For example,

* Example.* Matrix multiplication obeys some of
the algebraic laws you're familiar with. For example, matrix
multiplication is * associative*: If A, B, and C
are matrices and their dimensions are compatible for multiplication,
then

However, matrix multiplication is *not* *
commutative* in general. That is, for all matrices A, B.

One trivial way to get a counterexample is to let A be and let B be . Then is while is .

However, it's easy to come up with counterexamples even when and have the same dimensions. For example, consider the following matrices in :

Then

* Example.* If A is a matrix, the * transpose* of A is obtained by
swapping the rows and columns of A. For example,

Notice that the transpose of an matrix is an matrix.

* Example.* Consider the following matrices with
real entries:

(a) Compute .

(b) Compute .

* Example.* The * inverse* of
an matrix A is a matrix which satisfies

There is no such thing as matrix division in general, because some matrices do not have inverses. But if A has an inverse, you can simulate division by multiplying by . This is often useful in solving matrix equations.

For example, if

is a matrix, the inverse of A --- if there is one --- turns out to be

To show that this is the inverse of A, I have to check that and :

This proves that the formula gives the inverse of a matrix.

Here's an example for a matrix with real entries:

If , you can't use this formula; in fact, a
matrix which satisfies this condition does not have an inverse. A
matrix which does not have an inverse is called *
singular*.

For example,

* Example.* For what values of x is the following
real matrix singular?

The matrix is singular --- not invertible --- if

Solve for x:

The matrix is singular for and for .

* Example.* (* Solving a system
of equations)* Here's how to use inverses and matrix
multiplication to solve a system of equations. Suppose I want to
solve the following system over for x and y:

Write the equation in matrix form:

Multiply both sides by the inverse of the square matrix:

On the left, the square matrix and its inverse cancel, since they multiply to I. On the right,

Therefore,

The solution is , .

Copyright 2016 by Bruce Ikenaga