This entry is 5 because
(2)(3) + (0)(1) + (1)(−1) = 5.
Notice the sizes of the three matrices. A is 4×3, B is 3×2, and the product AB is 4×2.
We shall see in later chapters that this definition of matrix multiplication is exactly what is needed for applying matrices in our study of linear algebra.
It is an important consequence of this definition that:
AB 6=BA in general. That is, matrix multiplication is not ‘commutative’.
To see just how non-commutative matrix multiplication is, let’s look at some examples, starting with the two matrices A and B in the example above. The product AB is defined, but the product BA is not even defined. Since A is 4×3 and B is 3×2 it is not possible to multiply the matrices in the order BA.
Now consider the matrices A=
2 1 3
1 2 1
and B =
3 1 1 0 1 1
.
Both productsAB and BA are defined, but they are different sizes, so they cannot be equal. What sizes are they?
Activity 3.5 Answer the question just posed concerning the sizes of AB and BA.
Multiply the matrices to find the two product matrices, AB and BA.
Even if both products are defined and the same size, it is still generally true that AB6=BA.
Activity 3.6 Try this for any two 2×2 matrices. Write down two different matrices A and B and find the products AB and BA. For example, you could use
A=
1 2 3 4
B =
1 1 0 1
.
3.4 Matrix algebra
Matrices are useful because they provide a compact notation and we can do algebra with them.
For example, given a matrix equation such as
3A+ 2B = 2(B−A+C),
3
we can solve this for the matrixC using the rules of algebra. You must always bear in mind that to perform the operations, they must be defined. In this equation it is understood that all the matrices A, B and C are the same size, say m×n.
We list the rules of algebra satisfied by the operations of addition, scalar multiplication and matrix multiplication. The sizes of the matrices are dictated by the operations being defined.
A+B =B+A. Matrix addition is ‘commutative’.
This is easily shown to be true. We will carry out the proof as an example. The matrices A and B must be of the same size, say m×n for the operation to be defined, so both A+B and B+A are alsom×n matrices. They also have the same entries. The (i, j) entry ofA+B isaij+bij and the (i, j) entry ofB+A isbij+aij, butaij+bij =bij+aij by the properties of real numbers. So the matrices A+B and B+A are equal.
On the other hand, as we have seen, matrix multiplication is not commutative:
AB6=BA in general.
We have the following ‘associative’ laws:
(A+B) +C =A+ (B+C) λ(AB) = (λA)B =A(λB) (AB)C =A(BC)
These rules allow us to remove brackets. For example the last rule says that we will get the same result if we first multiplyAB and then multiply by C on the right, as we will if we first multiplyBC and then multiply by A on the left, so the choice is ours.
All these rules follow from the definitions of the operations in the same way as we showed the commutativity of addition. We need to know that the matrices on the left and on the right of the equals sign have the same size and that corresponding entries are equal. Only the associativity of multiplication presents any complications; it is tedious, but it can be done.
Activity 3.7 Think about these rules. What sizes are each of the matrices?
Write down the (i, j) entry for each of the matricesλ(AB) and (λA)(B) and prove that the matrices are equal.
3
3.4. Matrix algebra
Similarly, we have three ‘distributive’ laws:
A(B+C) =AB+AC (B+C)A =BA+CA λ(A+B) = λA+λB.
Why do we need both of the first two rules (which state that matrix multiplication distributes through addition)? Since matrix multiplication is not commutative, we cannot conclude the second distributive rule from the first; we have to prove it is true separately. All these statements can be proved from the definitions using the same technique as used earlier, but we will not take the time to do this here.
IfA is an m×n matrix, what is the result of A−A? We obtain an m×n matrix all of whose entries are 0. This is an ‘additive identity’: that is, it plays the same role for matrices as the number 0 does for numbers, in the sense that A+ 0 = 0 +A=A. There is a zero matrix of any size m×n.
Definition 3.6 (Zero matrix) A zero matrix, denoted 0, is anm×n matrix with all entries zero,
0 =
0 0 · · · 0 0 0 0 · · · 0 0 ... ... . .. ... ...
0 0 · · · 0 0
.
Then
A+ 0 =A A−A= 0
0A= 0, A0 = 0
where the sizes of the zero matrices above must be compatible with the size of the matrix A for the operations to be defined.
Activity 3.8 IfA is a 2×3 matrix, write down the zero matrix for each of the rules involving addition. What sizes are the zero matrices for the rules involving matrix multiplication?
What about matrix multiplication? Is there a ‘multiplicative identity’, which acts like the number 1 does for multiplication of numbers? The answer is ‘yes’.
Definition 3.7 (Identity matrix) The n×n identity matrix, denoted In or simplyI, is the diagonal matrix with aii = 1, 1≤i≤n,
I =
1 0 · · · 0 0 1 · · · 0 ... ... . .. ...
0 0 · · · 1
.
3
If A is anym×n matrix, then, AI =A and IA =A,
where it is understood that the identity matrix is the appropriate size for the product to be defined.
Activity 3.9 What size is the identity matrix ifA is m×n and IA=A?
Activity 3.10 You can apply these rules to solve the matrix equation, 3A+ 2B = 2(B−A+C)
for the matrix C. Do this; solve the equation for C, stating what rule of matrix algebra you are using for each step of the solution.
R You should now read sections 1.3 and 1.4 of the text A-H, working through the activities there. You will find the solution of the last activity in the text.