Using Matrices to Solve Linear Systems
It is not uncommon for a car dealership to have multiple stores across a geographic region. Consider the case of the US Auto Import chain with stores in Yonkers, Croton-on-Harmon, Saratoga Springs, Syracuse, and Ithaca. The May inventory is taken of four of its best sellers: the Toyota Avalon, the Dodge Durango, the Jaguar S-type, and the BMW 530i sedan. Table 6.1 shows how many of each model are in stock at each location.
The wholesale price of each model (that is, the price that US Auto Import paid for each model) is given in Table 6.2.
Example 1: Compute the value of the inventory for the month of May at each of the five US Auto Import stores.
For each store, multiply the number of each model by the wholesale cost of the model.
We will now use this example to illustrate the mathematical construct called a matrix. A matrix is a rectangular array of numbers. Ignoring the labels that are included to help read the tables, Table 6.1 has five rows and four columns. Rows are read horizontally and columns vertically. Table 6.2 has four rows and one column. The dimensions of a matrix are determined by the number of rows and the number of columns.
To enter a new matrix in TI 83 + /84, the keystrokes are 2nd x-1, left arrow, ENTER, and type the dimensions of the matrix. The dimensions of the matrix, 5 rows and 4 columns, are displayed on the top line of the screen as 5×4. The notation A5,4 indicates that the matrix has dimensions 5×4.
Notice the number in the lower left-hand corner of the screen. The entry in column 1 row 1 is 0. For this problem, row 1 corresponds to the inventory in Yonkers. Enter 25 and press e.
As 25 is entered into the calculator, Screen D shows the display. After the ENTER key is struck, the calculator moves to the second column of row 1 and indicates that the current entry is 0. Finish entering the inventory values for matrix A. The limitations of the screen size and the size of the data prevent you from seeing all the data at once, but you can use the left and right arrows to scroll across the screen to be sure you have entered the data correctly (Screen F).
Edit matrix B to be 4×1 (4 rows, 1 column), and enter the values for the wholesale costs in a similar manner. Quit, and return to the home screen. Multiply matrices [A] and [B] by typing 2nd x-1 ENTER x 2nd x-1 2 to get
These are the same values calculated in the solution to Example 1; they reflect the total value of the inventory for each store. This tells you the process in which multiplication of matrices occurs. Starting with row 1 of the left-hand factor, each number in row 1 is multiplied by a number from column 1 in the right-hand factor, and these products are added to give the result.
Matrix multiplication and the inverse of a matrix can be used to solve systems of linear equations. For instance, consider the following system:
This is the equivalent of the matrix equation
Observe that the first matrix contains the same numbers as the coefficients from the system. The second matrix contains the variables of the system. The third system contains the constants of the linear equations. When you perform the matrix multiplication on the left, you see the same terms as appear on the left side of the system of equations.
This matrix equation can be written as [A][X]= [B]. In Algebra 1, you would solve the equation ax = b by dividing both sides of the equation by a to get x = b/a. However, you can't divide matrices. You could also solve the equation by multiplying both sides of the equation by the multiplicative inverse of
The solution to the system of equations is the point (12, −7).
The matrix equation is
Multiply by the inverse of the coefficient matrix to get