Practice Makes Perfect Algebra - Carolyn Wheater [34]
The product of two matrices is found by repeating a process of multiplying one row by one column. We will look at the details of that process in the sections that follow, but before we begin that exploration, it is important to state the conditions under which matrix multiplication is possible.
Two matrices can be multiplied only if the number of elements in each row of the first matrix is equal to the number of elements in each column of the second. If and , we can multiply A · B because there are three elements of each row of A and three elements of each column of B. In other words, the number of columns in the first matrix must be equal to the number of rows in the second. Matrix A has two rows and three columns, and matrix B has three rows and four columns.
If we write the dimensions of matrix A and then the dimensions of matrix B, we can see a simple way to tell if the multiplication is possible(see Figure 13.5).
Figure 13.5 If dimensions match, the multiplication is possible.
If the dimensions do not match, the multiplication cannot be performed(see Figure 13.6). The need for matching dimensions makes the order of multiplication important. While we have seen that it is possible to multiply A · B, it is not possible to perform the multiplication B · A.
Figure 13.6 These numbers do not match, so the multiplication is not possible.
Looking at the dimensions can give us another piece of useful information as well. The remaining numbers tell us the dimension of the product matrix that will result. When we multiply A · B looking at dim(A) 2 × 3 and dim(B) 3 × 4 tells us not only that the multiplication is possible, but also that the product matrix will have dimension 2 × 4 as shown in Figure 13.7.
Figure 13.7 These numbers tell the dimension of the product matrix.
Multiplying a single row by a single column
To understand the process of matrix multiplication, we will focus first on a row matrix times a column matrix. Form a row matrix with the prices of the T-shirts and the sweatshirts in our earlier example, [12.99 15.99], and a column matrix with the total sales by the senior class, . The element 172 represents the number of T-shirts sold by the senior class, and the element 95 is the number of sweatshirts they sold.
In order to find the total amount that the senior class raised, we need to multiply $12.99 times 172 and multiply $15.99 times 95 and add the results together. The first element in the row is multiplied by the first element in the column, and then the second element in the row is multiplied by the second element in the column. These products are added to form the single element in the product matrix. The result of multiplying a 1 × 2 row matrix times a 2 × 1 column matrix is a 1 × 1 matrix.
If there are more elements in the row and the column—remember that the number of elements in the row must match the number of elements in the column—then there are additional products, but all are combined to produce a single element in the product matrix.
Multiplying a single row by a larger matrix
In our row-times-column example, we found the total amount of money raised by the senior class. We could repeat the exercise for each of the classes, but it would be simpler if we could multiply the row matrix containing prices [12.99 15.99] by the 2 × 4 matrix containing the numbers of T-shirts and sweatshirts sold by each class. Multiplying this 1 × 2 matrix by a 2 × 4 matrix should produce a 1 × 4 matrix, which logically would contain the fund-raising totals for each of the four classes.
To multiply a row matrix by a matrix with more than one column, multiply the row matrix times the first column of the larger matrix to produce the first element of the product matrix. Repeat the multiplication using the row