In this section we’re going to take a quick look at some of the properties of matrix arithmetic and of the transpose of a matrix.  As mentioned in the previous section most of the basic rules of real number arithmetic are still valid in matrix arithmetic.  However, there are a few that are no longer valid in matrix arithmetic as we’ll be seeing.

We’ve already seen one of the real number properties that doesn’t hold in matrix arithmetic.  If a and b are two real numbers then we know by the commutative law for multiplication of real numbers that  (i.e. (2)(3)=(3)(2)=6 ).  However, if A and B are two matrices such that AB is defined we saw an example in the previous section in which BA was not defined as well as an example in which BA was defined and yet .  In other words, we don’t have a commutative law for matrix multiplication.  Note that doesn’t mean that we’ll never have  for some matrices A and B, it is possible for this to happen (as we’ll see in the next section) we just can’t guarantee that this will happen if both AB and BA are defined.

Now, let’s take a quick look at the properties of real number arithmetic that are valid in matrix arithmetic.

Properties

With real number arithmetic we didn’t need both 4. and 5. since we’ve also got the commutative law for multiplication.  However, since we don’t have the commutative law for matrix multiplication we really do need both 4. and 5.  Also, properties 6.  9. are simply distributive or associative laws for dealing with scalar multiplication.

Now, let’s take a look at couple of other idea from real number arithmetic and see if they have equivalent ideas in matrix arithmetic.

We’ll start with the following idea.  From real number arithmetic we know that .  Or, in other words, if we multiply a number by 1 (one) doesn’t change the number.  The identity matrix will give the same result in matrix multiplication.  If A is an  matrix then we have,

Note that we really do need different identity matrices on each side of A that will depend upon the size of A.

Now, just like the identity matrix takes the place of the number 1 (one) in matrix multiplication, the zero matrix (denoted by 0 for a general matrix and 0 for a column/row matrix) will take the place of the number 0 (zero) in most of the matrix arithmetic.  Note that we said most of the matrix arithmetic.  There are a couple of properties involving 0 in real numbers that are not necessarily valid in matrix arithmetic.

Let’s first start with the properties that are still valid.

Zero Matrix Properties

Now, in real number arithmetic we know that if  and  then we must have  (sometimes called the cancellation law).  We also know that if  then we have  and/or  (sometimes called the zero factor property).  Neither of these properties of real number arithmetic are valid in general for matrix arithmetic.

We should be careful and not read too much into the results of the previous example.  The cancellation law will not be valid in general for matrix multiplication.  However, there are times when a variation of the cancellation law will be valid as we’ll see in the next section.

Now, again, we need to be careful.  There are times when we will have a variation of the zero factor property, however there will be no zero factor property for the multiplication of any two random matrices.

The next topic that we need to take a look at is that of powers of matrices.  At this point we’ll just work with positive exponents.  We’ll need the next section before we can deal with negative exponents.  Let’s start off with the following definitions.

We’ve also got several of the standard integer exponent properties that we are used to working with.

Properties of Matrix Exponents

We can also talk about plugging matrices into polynomials using the following definition.  If we have the polynomial,

and A is a square matrix then,

where the identity matrix on the constant term  has the same size as A.