MAT Solving Linear Systems Using Matrices and Row Operations


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1 MAT Solving Linear Systems Using Matrices and Row Operations A. Introduction to Matrices Identifying the Size and Entries of a Matrix B. The Augmented Matrix of a System of Equations Forming Augmented Matrices Writing the System Corresponding to an Augmented Matrix Mar 15 10:35 AM C. Solving a System Using Matrices Elementary Row Operations 1. Any two rows in a matrix can be interchanged. 2. The elements of any row can be multiplied by a nonzero constant. 3. Any two rows can be added together, and the sum used to replace one of the rows. In this section, we ll use these operations to triangularize the augmented matrix, employing a solution method known as Gaussian elimination. A matrix is said to be in triangular form when all of the entries below the diagonal are zero. Apr 20 6:58 AM 1
2 Solving Systems by Triangularizing the Augmented Matrix 1. Write the system as an augmented matrix. 2. Use row operations to obtain zeroes below the first diagonal entry. 3. Use row operations to obtain zeroes below the second diagonal entry. 4. Continue until the matrix is triangularized ( entries below diagonal are zero). 5. Divide to obtain a 1 in the last diagonal entry ( if it is nonzero), then convert to equation form and solve using back substitution. Note: At each stage, look for opportunities to simplify row entries using multiplication or division. Also, to begin the process any equation with an x coefficient of 1 can be made R1 by interchanging the equations. Apr 20 7:02 AM D. Inconsistent and Dependent Systems Apr 20 7:06 AM 2
3 E. Solving Applications Using Matrices TECHNOLOGY HIGHLIGHT Solving Systems Using Matrices and Calculating Technology From the first row, we get x = 3. From the second row, we get y = 1. From the third row, we get z = 1. So, the solution is ( 3, 1, 1). Apr 20 7:10 AM 776/8. Determine the size ( order) of each matrix and identify the third row and second column entry. If the matrix given is a square matrix, identify the diagonal entries. 3
4 776/9. Determine the size ( order) of each matrix and identify the third row and second column entry. If the matrix given is a square matrix, identify the diagonal entries. 776/10. Form the augmented matrix, then name the diagonal entries of the coefficient matrix. 4
5 776/12. Form the augmented matrix, then name the diagonal entries of the coefficient matrix. 777/15. Write the system of equations for each matrix. Then use back substitution to find its solution. 5
6 777/16. Write the system of equations for each matrix. Then use back substitution to find its solution. 777/22. Perform the indicated row operation( s) and write the new matrix. 6
7 777/24. Perform the indicated row operation( s) and write the new matrix. 777/26. What row operations would produce zeroes beneath the first entry in the diagonal? 7
8 777/27. What row operations would produce zeroes beneath the first entry in the diagonal? 777/32. Solve each system by triangularizing the augmented matrix and using back substitution. Simplify by clearing fractions or decimals before beginning. 8
9 777/36. Solve each system by triangularizing the augmented matrix and using back substitution. Simplify by clearing fractions or decimals before beginning. 777/38. Solve each system by triangularizing the augmented matrix and using backsubstitution. If the system is linearly dependent, give the solution in terms of a parameter. If the system has coincident dependence, answer in set notation as in Chapter 5. 9
10 778/41. Solve each system by triangularizing the augmented matrix and using back substitution. If the system is linearly dependent, give the solution in terms of a parameter. If the system has coincident dependence, answer in set notation as in Chapter /42. Solve each system by triangularizing the augmented matrix and using back substitution. If the system is linearly dependent, give the solution in terms of a parameter. If the system has coincident dependence, answer in set notation as in Chapter 5. 10
11 778/43. Solve each system by triangularizing the augmented matrix and using back substitution. If the system is linearly dependent, give the solution in terms of a parameter. If the system has coincident dependence, answer in set notation as in Chapter /44. Solve each system by triangularizing the augmented matrix and using back substitution. If the system is linearly dependent, give the solution in terms of a parameter. If the system has coincident dependence, answer in set notation as in Chapter 5. 11
12 778/45. Solve each system by triangularizing the augmented matrix and using back substitution. If the system is linearly dependent, give the solution in terms of a parameter. If the system has coincident dependence, answer in set notation as in Chapter 5. 12
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