Practice Makes Perfect Algebra - Carolyn Wheater [35]
The multiplication tells us that the senior class raised $3763.33, the junior class $2118.51, the sophomore class $2576.19, and the freshman class $737.46.
Multiplying matrices
The process of multiplying two larger matrices repeats these same steps, with each row of the first matrix producing a row of the product matrix. To multiply two matrices, begin by multiplying the first row of the first matrix by each column of the second matrix, placing the results in the first row of the product matrix. Repeat the process using each row of the first matrix, and place the results in the corresponding row of the product matrix.
As an example, we will multiply the 4 × 2 matrix by the 2 × 3 matrix .
A look at the dimensions of the matrices tells us that the multiplication is possible and that the product matrix will be 4 × 3(see Figure 13.8).
Figure 13.8
Focus on the first row.
Repeat for the second row.
Then the third row.
And finally for the bottom row.
Matrix multiplication is often useful in real-world situations. Suppose a small electronics firm ships TVs, VCRs, and DVDs to stores in New York, Chicago, and Los Angeles. The number of each item shipped to each city can be organized into the matrix . The company would want to record the cost of each item, the selling price, and the shipping cost. This would result in the matrix Multiplying these two matrices can provide the company with important information about its costs.
Care must be taken to assure that the order of the multiplication not only meets the dimension requirement, but also makes sense in terms of the quantities being multiplied. In our example, both matrices are 3 × 3, so the order of the multiplication is not obvious. If we look at the labels on our matrices, however, the order becomes clearer.
The first matrix has a row for each city and a column for each product. We can represent this as City × Product. The second has a row for each product and a column for each cost. This would be represented as Product × Cost. The order City × Product times Product × Cost provides a match, both in number and kind, between the center dimensions, and tells us that the product will be City × Cost. This means the product matrix will have a row for each city and a column for each cost. Attempting to multiply in the other order would be possible in terms of the size of the matrices but would make little sense when the categories of information were considered.
The identity matrix
When you multiply , you may note an interesting result. The product is identical to the second matrix. . If you explore a little, you’ll find that anytime you multiply by a square matrix with 1s on the diagonal and 0s everywhere else, it leaves the other matrix unchanged. It’s the matrix equivalent of multiplying a number by 1. A square matrix with 1s on the main diagonal and 0s elsewhere is called an identity matrix.
Identity matrices come in various sizes, but they’re always square. To multiply the matrix by an identity on the left, you’ll need a 2 × 2 identity.
To multiply the same matrix by an identity on the right, you’ll need a 4 × 4 identity.
If the matrix M is a square matrix, then for an identity matrix I of the same dimension as M, M × I = I × M = M.
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EXERCISE 13.4
For probs. 1 through 13, use the matrices
Determine whether each multiplication is possible. If it is possible, give the dimension of the product matrix.
1. A × B
2. B × A
3. A × C
4. C × A
5. B × C
6. C × B
7. B × D
8. D × B
9. B × E
10. E × B
11. D × E
12. E × D
13. A × D
Multiply, if possible.
20. A nutritionist prepares menus for three groups of patients: adults with diabetes, men with coronary disease, and nursing mothers. In planning meals, she chooses from the same group of foods, varying the selections and the portion sizes to meet the differing needs of the groups. For a dinner menu, she can choose from beef, chicken, scalloped potatoes, rice, broccoli, green beans, bread, butter, low-fat milk, brownies, and apples.