Practice Makes Perfect Algebra - Carolyn Wheater [32]
Clothing: 5 tennis, 2 golf, 1 basketball, 2 softball
Accessories: 3 golf, 1 volleyball, 5 softball
Books: 2 tennis, 12 golf, 1 basketball
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Addition and subtraction
The sporting goods store mentioned in the last section would certainly want to combine the information gathered on one day with information from other days to see total sales for a week, a month, a quarter, or a year. This could be accomplished by adding the individual numbers, but entering the information into matrices, as you did in the exercise, can simplify the process. With the aid of calculators or computers, the computation is streamlined, becoming one operation rather than many, but even if the work must be done manually, the matrix structure clarifies the task and helps to prevent errors.
Matrices to be added or subtracted must be of the same order. Only matrices with identical dimensions can be added or subtracted. If the sizes of the matrices are not the same, the addition or subtraction cannot be performed.
Logically, it is also important to consider what the matrices represent. It would make little sense to add a matrix showing sales of sporting goods to a matrix containing calorie counts. What could the total possibly represent? Even if the differences between the matrices are not so dramatic, care must be taken to assure that the calculation is sensible.
If two matrices have the same dimension, they can be added by simply adding the corresponding elements. In symbolic terms, if we want to add matrix A to matrix B and the matrices have the same dimension, we add a(1, 1) + b(1, 1), a(1, 2) + b(1, 2), and so on. If the matrices did not have the same dimension, some elements would not have partners, and it would be impossible to complete the addition properly.
To add the matrix to the matrix , we create a new matrix of the same dimension, and fill it with the sums of the corresponding elements.
The process of subtracting matrices is similar to that of adding matrices. Matrices must be of the same dimension, and corresponding elements are subtracted. To subtract the matrix from the matrix , form a new matrix of the same dimension and fill it with the differences of the corresponding elements.
Just as in standard arithmetic, order is significant in subtraction. You know that 7 − 3 ≠ 3 − 7. The first equals 4, while the second gives −4. Similarly, the result of is not the same as .
Changing the order of subtraction changed the sign of each element in the final matrix.
When you learned to subtract integers, you probably were taught to “add the opposite” or to “change the sign and add.” These rules told you, for example, that 7 −(−3) = 7 +(+3). When subtracting matrices, you can apply a similar rule.
and the difference can be found by subtracting the corresponding elements. If it is more convenient, however, the problem can be expressed as . Each element of the second matrix has been changed to its opposite, and you add instead of subtracting. Only the second matrix, the one following the minus sign, is changed.
or
When matrix addition or subtraction is used in applications, it is important to be certain that the matrices are organized in ways that assure that the operation is sensible. Attempting to add the matrix
to the matrix
would produce numbers with little meaning, since the categories appear in different orders in each matrix. One of the matrices should first be reorganized so that corresponding elements represent like quantities.
The need for matching dimensions becomes clearer when we see the matrices in context. If we tried to add the matrix
to the matrix
someone looking at the result would have no way to know that the Basketball column represented only 1 day of sales while the other columns represented 2 days.
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EXERCISE 13.2
Add, if possible.
Subtract, if possible.
11. A small investment club divides its funds among five stocks. The matrices below show the dividends each stock paid during each of the four quarters last