## 3.2 Matrix Multiplication

(This topic is also in Section 3.2 in Finite Mathematics and Finite Mathematics & Applied Calculus)

Some On-line Resources for This Topic:

In the preceding tutorial we learned how to add matrices and multiply matrices by scalars (real numbers). For example,

 2 -3 1 0 -1 3
+  9 -5 0 13 -1 3
=  11 -8 1 13 -2 6
4  1 3 5 -1 -8 10 -7 -5 13
=  4 12 20 -4 -32 40 -28 -20 52
Each entry multiplied by 4

Q What about matrix multiplication? Do we multiply matrices as follows?

 0 -1 -1 2
 3 -2 1 0
=  0 2 -1 0
?

A No, that is not the way we multiply matrices, since multiplying matrices in this way is not as useful as the "correct" way.

Q OK. What is the "correct way"?
A Since the correct way may seem a little strange at first, let us do it step-by-step.

A Row Times a Column

If A is a row matrix, and B is a column matrix, then their product AB is a 11 matrix. The length of the row in A must match the length of the column in B for the product to be defined. To find the product, multiply each entry in A (going from left to right) by the corresponding entry in B (going from top to bottom) and then add the results.

Examples

 [ 3     -2 ]
 4 5
= [34 + (-2) 5] = [2]     Product is a 11 matrix
 [ 3     -1     -2 ]
 4 3 1
=

Q How can we apply this strange method of multiplying rows by columns?
A Here is an application: Suppose you sell 3 T-shirts at \$10 each, 4 hats at \$15 each, and 1 pair of shorts at \$20. Then your total revenue is

 [ 10     15     20 ]
 3 4 1
= [103 + 154 + 120] = [110]
PriceQuantity =Revenue

The Product of Two Matrices: General Case

In general, we can take the product AB only if the number of columns of A equals the number of rows of B (so that we can multiply the rows of A by the columns of B as above). The product AB is then obtained as follows:

• To obtain the 1,1 entry of AB, multiply Row 1 of A by Column 1 of B.
• To obtain the 1,2 entry of AB, multiply Row 1 of A by Column 2 of B.
• To obtain the 1,3 entry of AB, multiply Row 1 of A by Column 3 of B.
. . .
• To obtain the 2,1 entry of AB, multiply Row 2 of A by Column 1 of B.
• To obtain the 2,2 entry of AB, multiply Row 2 of A by Column 1 of B.
and so on. In general,
To obtain the i,j entry of AB, multiply Row i of A by Column j of B.

Note: The product AB has as many rows as A and as many columns as B.

Examples

 [ 2     0     -1   1 ]
 1 1 0 2
 5 1 -1 0
 -7 0 1 0
=
 4
 11
 -15
14 43 13
 [ 3     -1     -2 ]
 4 2 0 3 0 2 1 1 0
=
 2 0 -1 1 1 2 0 1
 1 1 0 2
 5 1 -1 0
 -7 0 1 0
=
 4 5
 11 7
 -15 -7
24 43 23
 3 -1 -2 2 -2 -1
 4 2 0 3 0 2 1 1 0
=
 2 -1 1 3
 1 1 0 -1
=
 1 1 0 -1
 2 -1 1 3
=

If you compare the answers to the above two examples, you will see that they are different. In other words,
A.B B.A
Matrix multiplcation is not commutative.
 1 0 0 1 0 -1 0 1 -1
 1 2 3 2 3 4 2 4 6
=

Q Now give me an application for multiplying matrices with more than one row and/or column.
A Here is an application that extends the application above: Suppose you make the following sales:

Monday: 3 T-shirts at \$10 each, 4 hats at \$15 each, and 1 pair of shorts at \$20.
Tuesday: 4 T-shirts at \$10 each, 2 hats at \$15 each, and 3 pairs of shorts at \$20.
Then your total revenue for the two days is
 [ 10     15     20 ]
 3 4 4 2 1 3
= [110   130]
PriceQuantity =Revenue

There are tons of other applications in Exercise Set 3.2 of Finite Mathematics and Finite Mathematics & Applied Calculus Also see theon-line review exercises.

Identity Matrices

An identity matrix is a square matrix, called I, that has ones down the leading diagonal (see the examples below) and zeros everywhere else.

Here are some identity matrices.

I = [ 1 ]     11 identity matrix
 I = 1 0 0 1
22 identity matrix
 I = 1 0 0 0 1 0 0 0 1
33 identity matrix
 I = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
44 identity matrix

Note The leading diagonal of a square matrix is the diagonal from the top left to the bottom right. It consists of the entries a11, a22, a33, ..., ann.

Q Why are these matrices called "identity matrices"?
A Let us mutliply an identity matrix by some other matrix and see what we get:

 1 0 0 0 1 0 0 0 1
 1 2 3 4 5 6 7 8 9
=

In other words, is seems that that, if A is any square matrix, then

I.A   =

In other words, multiplying a matrix by I leaves it unchanged. This is analogous to multiplying a number, a, by 1. You should now try multiplying a 2 2 matrix by the 2 2 identity matrix. You will once again end up with the matrix you started with.

Q What happens if you multiply them the other way around? In other words, what is A.I? (Remember that matrix multiplcation is not commutative)
A Try it out, and you will find once again that the result is A. In other words,

A.I = I.A = A

for every square matrix A.

Here is a list of rules that summarize the multiplicative properties of matrices.

Properties of Matrix Multiplication

If A, B and C are any matrices, and if I is an identity matrix, then the following hold, whenever the dimensions of the matrices are such that the products are defined.

 A(BC) = (AB)C Associative Law A.I = I.A = Multiplicative Identity Law A(B + C) = AB + AC Left Distributive Law (A + B)C = Right Distributive Law A.0 = 0.A = 0 Multiplication by Zero

Now try some of the exercises in Section 3.1 of Finite Mathematics or Finite Mathematics and Applied Calculus

Last Updated: May, 2006