MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 9: Systems of Equations and Matrices 9.1 Systems of Equations in Two Variables 9.2 Systems of Equations in Three Variables 9.3 Matrices and Systems of Equations Technology, Matrices, and Systems of Equations See the Matrices section (see image below) of the tutorial at http://media.pearsoncmg.com/aw/aw_mml_shared_1/gc_tutorial/start.html The TI Calculator tutorials on the Important Links webpage include instructions for Matrix Operations with the TI calculator. This tutorial is available at http://cfcc.edu/faculty/cmoore/ti83matrixo per.htm This is a very good (9.5 minutes) video on YouTube about matrix operations. Solve system of 3 equations (1) using matrix inverse multiplication and (2) using augmented matrix solution (rref). http://www.youtube.com/watch?v=7e1uywdtonc&feature=related 9.3 Matrices and Systems of Equations Solve systems of equations using matrices. Matrices The system can be expressed as where we have omitted the variables and replaced the equals signs with a vertical line. Matrices A rectangular array of numbers such as is called a matrix (plural, matrices). The matrix is an augmented matrix because it contains not only the coefficients but also the constant terms. The matrix is called the coefficient matrix. 1
Matrices continued The rows of a matrix are horizontal. The columns of a matrix are vertical. The matrix shown has 2 rows and 3 columns. A matrix with m rows and n columns is said to be of order m by n. When m = n the matrix is said to be square. Gaussian Elimination with Matrices Row Equivalent Operations 1. Interchange any two rows. 2. Multiply each entry in a row by the same nonzero constant. 3. Add a nonzero multiple of one row to another row. We can use the operations above on an augmented matrix to solve the system. Example Solve the following system: Example continued New row 1 = row 2 New row 2 = row 1 First, we write the augmented matrix, writing 0 for the missing y term in the last equation. We multiply the first row by 2 and add it to the second row. We also multiply the first row by 4 and add it to the third row. Our goal is to find a row equivalent matrix of the form. 2
Example continued We multiply the second row by 1/5 to get a 1 in the second row, second column. Example continued We back substitute 3 for z in equation (2) and solve for y. We multiply the second row by 12 and add it to the third row. Next, we back substitute 1 for y and 3 for z in equation (1) and solve for x. Now, we can write the system of equations that corresponds to the last matrix above: The triple (2, 1, 3) checks in the original system of equations, so it is the solution. Row Echelon Form 1. If a row does not consist entirely of 0 s, then the first nonzero element in the row is a 1 (called a leading 1). 2. For any two successive nonzero rows, the leading 1 in the lower row is farther to the right than the leading 1 in the higher row. 3. All the rows consisting entirely of 0 s are at the bottom of the matrix. If a fourth property is also satisfied, a matrix is said to be in reduced row echelon form: 4. Each column that contains a leading 1 has 0 s everywhere else. Example Which of the following matrices are in row echelon form? a) b) c) d) Matrices (a) and (d) satisfy the row echelon criteria. In (b) the first nonzero element is not 1. In (c), the row consisting entirely of 0 s is not at the bottom of the matrix. 3
Gauss Jordan Elimination We perform row equivalent operations on a matrix to obtain a row equivalent matrix in row echelon form. We continue to apply these operations until we have a matrix in reduced row echelon form. Example: Use Gauss Jordan elimination to solve the system of equations from the previous example; we had obtained the matrix Gauss Jordan Elimination continued We continue to perform row equivalent operations until we have a matrix in reduced row echelon form. Next, we multiply the second row by 3 and add it to the first row. Gauss Jordan Elimination continued Special Systems When a row consists entirely of 0 s, the equations are dependent. For example, in the matrix Writing the system of equations that corresponds to this matrix, we have the system is equivalent to We can actually read the solution, (2, 1, 3), directly from the last column of the reduced row echelon matrix. 4
Special Systems 754/1 through 6. Determine the order (size) of the matrix. When we obtain a row whose only nonzero entry occurs in the last column, we have an inconsistent system of equations. For example, in the matrix the last row corresponds to the false equation 0 = 9, so we know the original system has no solution. 754/8. Write the augmented matrix for the system of equations. 3x + 2y = 8 and 2x 3y = 15. 754/10. Write the augmented matrix for the system of equations. x + y z = 7 3y + 2z = 1 2x 5y = 6 754/14. Write the system of equations that corresponds to the augmented matrix. 5
754/22. Solve the system of equations using Gaussian elimination or Gauss Jordan elimination. Use a graphing calculator to check your answer. x 4y = 9 and 2x + 5y = 5 754/34. Solve the system of equations using Gaussian elimination or Gauss Jordan elimination. Use a graphing calculator to check your answer. x + y 3z = 4 4x + 5y + z = 1 2x + 3y + 7z = 7 754/38. Solve the system of equations using Gaussian elimination or Gauss Jordan elimination. Use a graphing calculator to check your answer. a b + c = 3 2a + b 3c = 5 4a + b c = 11 755/43. Borrowing. Gonzalez Manufacturing borrowed $ 30,000 to buy a new piece of equipment. Part of the money was borrowed at 8%, part at 10%, and part at 12%. The annual interest was $ 3040, and the total amount borrowed at 8% and 10% was twice the amount borrowed at 12%. How much was borrowed at each rate? 6