MULTIPLYING MATRICES

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Before studying this web exercise, you may want to review Introduction to Matrices and Basic Arithmetic with Matrices .

Matrix multiplication is very different from matrix addition and subtraction.
You do not multiply corresponding entries; in particular, [2 3] times [4 5] is NOT NOT NOT [8 15] !
Indeed, we'll see that these two matrices aren't even "compatible" for matrix multiplication.

At first glance, the definition of matrix multiplication may seem strange and complicated.
However, it is defined in a way that makes it perfect for working with systems of equations.
The example below may help you to understand why multiplication of matrices is defined the way it is.

The students in a large high school (grades 9 through 12) get there in a variety of ways: by bike, by bus, and by car.
The percentage of students using different modes of transportation is summarized on the left below.
For example, 25% of 9^th grade students travel to school by bike.
The total number of male and female students in each grade is summarized in the table on the top right.
For example, there are 110 male 9^th grade students.

					GENDER	Male	Female
					9^th	110	105
					10^th	100	95
					11^th	95	90
					12^th	85	80
MODES OF TRANSPORTATION	9^th	10^th	11^th	12^th
bike	25%	20%	15%	10%		0.25(110) + 0.20(100) + 0.15(95) + 0.10(85) = 70	0.25(105) + 0.20(95) + 0.15(90) + 0.10(80) = 67
bus	55%	65%	55%	40%		0.55(110) + 0.65(100) + 0.55(95) + 0.40(85) = 212	0.55(105) + 0.65(95) + 0.55(90) + 0.40(80) = 201
car	20%	15%	30%	50%		0.20(110) + 0.15(100) + 0.30(95) + 0.50(85) = 108	0.20(105) + 0.15(95) + 0.30(90) + 0.50(80) = 102

The following observations are critical:

The column labels of the "Modes of Transportation" matrix (9^th, 10^th, etc.)
are the same as the row labels of the "Gender" matrix.
In particular, there are the same number of columns in the "Modes" matrix as there are rows in the "Gender" matrix.
Look at the green entries above:
The "bike" percentages (a row in the "Modes" matrix)
combine with the "male" numbers (a column in the "Gender" matrix)
to give the total number of males who bike to school.
The computation is a SUM of the PRODUCTS of corresponding entries.
Look at the purple entries above:
The "car" percentages (a row in the "Modes" matrix)
combine with the "female" numbers (a column in the "Gender" matrix)
to give the total number of females who go to school by car.
The computation is a SUM of the PRODUCTS of corresponding entries.

Now strip away the labels, record the percentages as decimals, and suppress the computations.
Put the "Modes" matrix in green, the "Gender" matrix in purple.
The product of these two matrices is shown in white.

				110	105
				100	95
				95	90
				85	80
0.25	0.20	0.15	0.10	row 1 green column 1 purple 70	row 1 green column 2 purple 67
0.55	0.65	0.55	0.40	row 2 green column 1 purple 212	row 2 green column 2 purple 201
0.20	0.15	0.30	0.50	row 3 green column 1 purple 108	row 3 green column 2 purple 102

Most conventionally, here is the typical display of this product of matrices:

[ 0.250.200.150.10 0.550.650.550.40 0.200.150.300.50 ] [ 110105 10095 9590 8580 ] = [ 7067 212201 108102 ]

Thus, we have:

DEFINITION: Matrix Multiplication

Suppose that A is an m×n matrix and B is an n×p matrix.
(In particular, the number of columns in A is the same as the number of rows in B .)

Then, the product P:=AB is defined, and has size m×p .
To find pij (the element in row i and column j of the matrix P ) :

take row i from matrix A ;
take column j from matrix B ;
form the sum of the products of corresponding entries.

That is, if the numbers in the row from A (left-to-right) are a1 ,a2, ...,an ,
and the numbers in the column from B (top-to-bottom) are b1 ,b2, ...,bn ,
then: pij =a1 b1 +a2 b2 +...+a nb n

Notice how easy it is to organize systems of equations in matrix form with this definition!
Check that the system 2x-3y =5
x+7y= 4
has the following matrix representation: [ 2-3 17 ] [ x y ] = [ 54 ]
Then, after additional tools for working with matrices are developed, the system can be easily solved using matrix methods.

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.

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