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• Matrix Algebra Tool
• Free Mac Software for Matrix Algebra

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	Corresponding entries added
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

?

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]
Price		Quantity	=	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.

[ 10     15     20 ]		3     4   4 2 1 3	=	[110   130]
Price		Quantity	=	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, ..., a_nn.

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

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.

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