Practice Makes Perfect Algebra - Carolyn Wheater [33]
12. The matrices below show the prices at which the investment club bought its stocks and the prices at which it sold them at the end of its investment term. Use matrix subtraction to determine the profit or loss on each stock.
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Scalar multiplication
In matrix arithmetic, there are two types of multiplication, scalar multiplication and matrix multiplication. As the name suggests, the latter involves the multiplication of a matrix by a matrix, and we will consider that in the next section. Scalar multiplication, on the other hand, is the multiplication of a single number times a matrix. The single number is called a scalar.
In the scalar multiplication , multiplying the 2 × 3 matrix by the scalar 3 has the effect of adding three copies of the matrix.
Because scalar multiplication represents repeated addition, we can easily anticipate the end result of the process.
Focusing on the beginning and end of this process allows us to find the common shortcut for scalar multiplication.
To multiply a matrix by a scalar, multiply each element of the matrix by the scalar.
The rule for scalar multiplication may remind you of the process you learned as the distributive law. While the scalar certainly seems to be distributed over the matrix, there is a significant difference between the two ideas. The distributive property distributes multiplication over addition(or subtraction), assuring us that c(a + b) = ca + cb. In scalar multiplication, the elements of the matrix are not added to one another, either before or after the multiplication.
When scalar multiplication is combined with addition and subtraction, the familiar order of operations will apply. First perform any scalar multiplication, then add or subtract from left to right. As always, matrices must have the same dimension if addition or subtraction is to be performed.
Applications to coordinate geometry
The name scalar multiplication comes from the fact that the single number multiplying the matrix represents a scale factor, an indication of a proportional change in the size. This root meaning of the term is simplest to see when we consider a figure in a coordinate plane represented by a matrix of its coordinates as shown in Figure 13.3.
Figure 13.3 The vertices of the triangle can be represented in a matrix.
If the triangle shown on the grid at the left is represented by the matrix and we multiply that matrix by a scalar factor of 2, we produce a new matrix.
If we graph the points represented by this new matrix, we find that they form the vertices of a triangle similar to the original, but with sides twice as long. Each vertex of the image triangle is twice as far from the origin as the corresponding vertex of the original triangle as shown in Figure 13.4.
Figure 13.4 The new triangle is twice the size.
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EXERCISE 13.3
Multiply.
Perform the indicated operations, if possible.
Graph the points represented by matrix A. Then perform the indicated scalar multiplication and graph the points represented by the answer matrix.
14. The four classes in a high school compete in a fund-raising event in which they sell T-shirts for $12.99 each. Seniors sold 172 shirts, juniors 88, sophomores 106, and freshmen 42. Organize the sales numbers into a matrix, and use scalar multiplication to find the amount of money raised by each class.
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Matrix multiplication
When a matrix is multiplied by a scalar, every element of the matrix is multiplied by that same value. While this is useful in some situations, many times the necessary calculations are more complicated.
In the last exercise of the previous section, you created a small matrix and multiplied it by the scalar $12.99. Imagine, however, that the classes sold both T-shirts and sweatshirts. For each class, you would need to record both the number of T-shirts sold and the number of sweatshirts sold. This increases the size of your matrix, but also introduces another problem. It is unlikely