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  However, there are a few that are no longer valid in matrix arithmetic as we’ll be seeing.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

In the following set of properties a and b are scalars and A, B, and C are matrices.  We’ll assume that the size of the matrices in each property are such that the operation in that property is defined.                                   Commutative law for addition              Associative law for addition                           Associative law for multiplication                     Left distributive law                      Right distributive law

 

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 ideas 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.

 

Example 1  Consider the following matrix.                                                                Then,

 

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

In the following properties A is a matrix and 0 is the zero matrix sized appropriately for the indicated operation to be valid.          and

 

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.

 

Example 2  Consider the following three matrices.                              We’ll leave it to you to verify that,                                                          Clearly  and just as clearly  and yet we do have .  So, at least in this case, the cancellation law does not hold.

 

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.

 

Example 3  Consider the following two matrices.                                            We’ll leave it to you to verify that,                                                                   So, we’ve got  despite the fact that  and .  So, in this case the zero factor property does not hold in this case.

 

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.

 

Definition 1 If A is a square matrix then,

 

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

 

Properties of Matrix Exponents

If A is a square matrix and n and m are integers then,

 

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.

 

Example 4  Evaluate each of the following for the given matrix.   (a)   (b)   (c)  where   Solution (a) There really isn’t much to do with this problem.  We’ll leave it to you to verify the multiplication here.                                             (b) In this case we may as well take advantage of the fact that we’ve got the result from the first part already.  Again, we’ll leave it to you to verify the multiplication.                                    (c) In this case we’ll need the result from the second part.  Outside of that there really isn’t much to do here.

 

The last topic in this section that we need to take care of is some quick properties of the transpose of a matrix.

 

Properties of the Transpose

If A and B are matrices whose sizes are such that the given operations are defined and c is any scalar then,

 

The first three of these properties should be fairly obvious from the definition of the transpose.  The fourth is a little trickier to see, but isn’t that bad to verify.

 

Proof of #4 : We know that the entry in the i^th row and j^th column of AB is given by,

We also know that the entry in the i^th row and j^th column of  is found simply by interchanging the subscripts i and j and so it is,

 

Now, let’s denote the entries of  and  as  and  respectively.  Again, based on the definition of the transpose we also know that,

 

and so from this we see that  and .  Finally, the entry in the i^th row and j^th column of  is given by,

 

Now, plug in for  and  and we get that,

So, just what have we done here?  We’ve managed to show that the entry in the i^th row and j^th column of  is equal to the entry in the i^th row and j^th column of .  Therefore, since each of the entries are equal the matrices must also be equal.

 

Note that #4 can be naturally extended to more than two matrices.  For example,