One of the biggest impediments that some people have in learning about matrices for the first time is trying to take everything that they know about arithmetic of real numbers and translate that over to matrices.  As you will eventually see much of what you know about arithmetic of real numbers will also be true here, but there are also a few ideas/facts that will no longer hold here.  To make matters worse there are some rules of arithmetic of real numbers that will work occasionally with matrices but won’t work in general.  So, keep this in mind as you go through the next couple of sections and don’t be too surprised when something doesn’t quite work out as you expect it to.

 

This section is devoted mostly to developing the arithmetic of matrices as well as introducing a couple of operations on matrices that don’t really have an equivalent operation in real numbers.  We will see some of the differences between arithmetic of real numbers and matrices mentioned above in this section.  We will also see more of them in the next section when we delve into the properties of matrix arithmetic in more detail.

 

Okay, let’s start off matrix arithmetic by defining just what we mean when we say that two matrices are equal.

 

Definition 1 If A and B are both  matrices then we say that A = B provided corresponding entries from each matrix are equal.  Or in other words, A = B provided  for all i and j.   Matrices of different sizes cannot be equal.

 

Example 1  Consider the following matrices.                          For these matrices we have that  and  since they are different sizes and so can’t be equal.  The fact that C is essentially the first column of both A and B is not important to determining equality in this case.  The size of the two matrices is the first thing we should look at in determining equality.   Next, A = B provided we have .  If  then we will have .

 

Next we need to move on to addition and subtraction of two matrices.

 

Definition 2 If A and B are both  matrices then  is a new  matrix that is found by adding/subtracting corresponding entries from each matrix.  Or in other words,                                                               Matrices of different sizes cannot be added or subtracted.

 

Example 2  For the following matrices perform the indicated operation, if possible.            (a)   (b)   (c)     Solution (a) Both A and B are the same size and so we know the addition can be done in this case. Once we know the addition can be done there really isn’t all that much to do here other than to just add the corresponding entries here to get the results. .                                                 (b) Again, since A and B are the same size we can do the difference and as like the previous part there really isn’t all that much to do.  All that we need to be careful with is the order.  Just like with real number arithmetic  is different from .  So, in this case we’ll subtract the entries of A from the entries of B.                                                     (c) In this case because A and C are different sizes the addition can’t be done.  Likewise, , , . , and  can’t be done for the same reason.

 

We now need to move into multiplication involving matrices.  However, there are actually two kinds of multiplication to look at : Scalar Multiplication and Matrix Multiplication.  Let’s start with scalar multiplication.

 

Definition 3 If A is any matrix and c is any number then the product (or scalar multiple), cA, is a new matrix of the same size as A and it’s entries are found by multiplying the original entries of A by c.  In other words  for all i and j.

 

Note that in the field of Linear Algebra a number is often called a scalar and hence the name scalar multiple since we are multiplying a matrix by a scalar (number).  From this point on we will generally call numbers scalars.

 

Before doing an example we need to get another quick definition out of the way.  If  are all matrices of the same size and  are scalars then the linear combination of  with coefficients  is,

 

This may seem like a silly thing to define but we’ll be using linear combination in quite a few places in this class and so we need to get used to seeing them.

 

Example 3  Given the matrices                            compute . Solution So, we’re really being asked to compute a linear combination here.  We’ll do that by first computing the scalar multiplies and the performing the addition and subtraction.  Note as well that in the case of the third scalar multiple we are going to consider the scalar to be a positive  and leave the minus sign out in front of the matrix.  Here is the work for this problem.

 

We now need to move into matrix multiplication, however before we do the general case let’s look at a special case first since this will help with the general case.

 

Suppose that we have the following two matrices,

So, a is a row matrix and b is a column matrix and they have the same number of entries.  Then the product of a and b is defined to be,

 

It is important to note that this product can only be done if a and b have the same number of entries.  If they have a different number of entries then this product is not defined.

 

Example 4  Compute ab given that,                                            Solution There is not really a whole lot to do here other than use the definition given above.

 

Now let’s move onto general matrix multiplication.

 

 

Definition 4 If A is an  matrix and B is a  matrix then the product (or matrix multiplication) is a new matrix with size  whose ijth entry is found by multiplying row i of A times column j of B.

 

So, just like with addition and subtraction, we need to be careful with the sizes of the two matrices we’re dealing with.  However, with multiplication we need to be a little more careful.  This definition tells us that the product AB is only defined if A (i.e. the first matrix listed in the product) has the same number of columns as B (i.e. the second matrix listed in the product) has rows.  If the number of columns of the first matrix listed is not the same as the number of rows of the second matrix listed then the product is not defined.

 

An easy way to check that a product is defined is to write down the two matrices in the order that we want to multiply them and underneath them write down the sizes as shown below.

If the two inner numbers are equal then the product is defined and the size of the product will be given by the outside numbers.

 

Example 5  Compute  and  for the following two matrices, if possible.                                    Solution Okay, let’s first do .  Here are the sizes for A and C.                                                          So, the two inner numbers (4 and 4) are the same and so the multiplication can be done and we can see that the new size of the matrix is .  Now, let’s actually do the multiplication.  We’ll go through the first couple of entries in the product in detail and then do the remaining entries a little quicker.   To get the number in the first row and first column of AC we’ll multiply the first row of A by the first column of B as follows,                                            If we next want the entry in the first row and second column of AC we’ll multiply the first row of A by the second column of B as follows,                                           Okay, at this point, let’s stop and insert these into the product so we can make sure that we’ve got our bearings.  Here’s the product so far,                                As we can see we’ve got four entries left to compute.  For these we’ll give the row and column multiplications but leave it to you to make sure we used the correct row/column and put the result in the correct place.  Here’s the remaining work.                                          Here is the completed product.                                Now let’s do CA.  Here are the sizes for this product.                                                          Okay, in this case the two inner numbers (3 and 2) are NOT the same and so this product can’t be done.

 

So, with this example we’ve now run across the first real difference between real number arithmetic and matrix arithmetic.  When dealing with real numbers the order in which we write a product doesn’t affect the actual result.  For instance (2)(3)=6 and (3)(2)=6.  We can flip the order and we get the same answer.  With matrices however, we will have to be very careful and pay attention to the order in which the product is written down.  As this example has shown the product AC could be computed while the product CA in not defined.

 

Now, do not take the previous example and assume that all products will work that way.  It is possible for both AC and CA to be defined as we’ll see in the next example.

 

Example 6  Compute BD and DB for the given matrices, if possible.                                    Solution First, notice that both of these matrices are  matrices and so both BD and DB are defined.  Again, it’s worth pointing out that this example differs from the previous example in that both the products are defined in this example rather than only one being defined as in the previous example.  Also note that in both cases the product will be a new  matrix.   In this example we’re going to leave the work of verifying the products to you.  It is good practice so you should try and verify at least one of the following products.

 

This example leads us to yet another difference (although it’s related to the first) between real number arithmetic and matrix arithmetic.  In this example both BD and DB were defined.  Notice however that the products were definitely not the same.  There is nothing wrong with this so don’t get excited about it when it does happen.  Note however that this doesn’t mean that the two products will never be the same.  It is possible for them to be the same and we’ll see at least one case where the two products are the same in a couple of sections.

 

For the sake of completeness if A is an  matrix and B is a  matrix then the entry in the i^th row and j^th column of AB is given by the following formula,

This formula can be useful on occasion, but is really used mostly in proofs and computer programs that compute the product of matrices.

 

On occasion it can be convenient to know a single row or a single column from a product and not the whole product itself.  The following theorem tells us how to get our hands on just that.

 

Theorem 1  Assuming that A and B are appropriately sized so that AB is defined then, The ith row of AB is given by the matrix product : [ith row of A]B. The jth column of AB is given by the matrix product : A[jth column of B].

 

Example 7  Compute the second row and third column of AC given the following matrices.   Solution These are the matrices from Example 5 and so we can verify the results of using this fact once we’re done.   Let’s find the second row first.  So, according to the fact this means we need to multiply the second row of A by C.  Here is that work.                                 Sure enough, this is the correct second row of the product AC.   Next, let’s use the fact to get the third column.  This means that we’ll need to multiply A by the third column of C.  Here is that work.                                                  And sure enough, this also gives us the correct answer.

 

We can use this fact about how to get individual rows or columns of a product as well as the idea of a partitioned matrix that we saw in the previous sections to derive a couple of new ways to find the product of two matrices.

 

Let’s start by assuming we’ve got two matrices A (size  ) and B (size  ) so we know the product AB is defined. 

 

Now, for the first new way of finding the product let’s partition A into its row matrices as follows,

Now, from the fact we know that the i^th row of AB is [i^th row of A]B, or .  Using this idea the product AB can then be written as a new partitioned matrix as follows.

 

For the second new way of finding the product we’ll partition B into its column matrices as,

We can then use the fact that t he j^th column of AB is given by A[j^th column of B] and so the product AB can be written as a new partitioned matrix as follows.

 

Example 8  Use both of the new methods for computing products to find AC for the following matrices.                                    Solution So, once again we know the answer to this so we can use it to check our results against the answer from Example 5.   First, let’s use the row matrices of A.  Here are the two row matrices of A.                                and here are the rows of the product.                           Putting these together gives,                                                 and this is the correct answer.   Now let’s compute the product using columns.  Here are the three column matrices for C.                                          Here are the columns of the product.                                                                                                                                 Putting all this together as follows gives the correct answer.

 

We can also write certain kinds of matrix products as a linear combination of column matrices.  Consider A an  matrix and x a  column matrix.  We can easily compute this product directly as follows,

 

Now, using matrix addition we can write the resultant  matrix as follows,

Now, each of the p column matrices on the right above can also be rewritten as a scalar multiple as follows.

Finally, the column matrices that are multiplied by the  ’s are nothing more than the column matrices of A.  So, putting all this together gives us,

where  are the column matrices of A.  Written in this matter we can see that  can be written as the linear combination of the column matrices of A, , with the entries of , , as coefficients.

 

Example 9  Compute  directly and as a linear combination for the following matrices.                                      Solution We’ll leave it to you to verify that the direct computation of the product gives,                                           Here is the linear combination method of computing the product.                                              This is the same result that we got by the direct computation.

 

Matrix multiplication also gives us a very nice and compact way of writing systems of equations.  In fact we even saw most of it as we introduced the above idea.  Let’s start out with a general system of  n equations and m unknowns.

Now, instead of thinking of these as a set of equations let’s think of each side as a vector of size  as follows,

In the work above we saw that the left side of this can be written as the following matrix product,

If we now denote the coefficient matrix by A, the column matrix containing the unknowns by x and the column matrix containing the  ’s by b. we can write the system in the following matrix form,

In many of the section to follow we’ll write general systems of equations as  given its compact nature in order to save space.

 

Now that we’ve gotten the basics of matrix arithmetic out of the way we need to introduce a couple of matrix operations that don’t really have any equivalent operations with real numbers.

 

Definition 5 If A is an  matrix then the transpose of A, denoted by , is an  matrix that is obtained by interchanging the rows and columns of A.  So, the first row of  is the first column of A, the second row of  is the second column of A, etc.  Likewise, the first column of  is the first row of A, the second column of  is the second row of A, etc.

 

On occasion you’ll see the transpose defined as follows,

Notice the difference in the subscripts.  Under this definition, the entry in the i^th row and j^th column of A will be in the j^th row and i^th column of . 

 

Notice that these two definitions are really the same definition, they just don’t look like they are the same at first glance.

 

Definition 6 If A is a square matrix of size  then the trace of A, denoted by tr(A), is the sum of the entries on main diagonal.  Or,                                                       If A is not square then the trace is not defined.

 

Example 10  Determine the transpose and trace (if it is defined) for each of the following matrices.                    Solution There really isn’t all that much to do here other than to go through the definitions.  Note as well that the trace will only not be defined for A and C since these matrices are not square.

 

In the previous example note that  and that .  In these cases the matrix is called symmetric.  So, in the previous example D and E are symmetric while A, B, and C, are not symmetric.