sec3.2.pptx chapter 3 for math for business

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1. Section 2 Systems of Linear Equations and Augmented Matrices Chapter 3 Systems of Linear Equations; Matrices

2. Matrices Solving systems of equations using elimination by addition involves manipulating the coefficients of the variables and the constant terms. The process is more efficient using a mathematical form called a matrix. A matrix is a rectangular array of numbers written within brackets. Each number in a matrix is called an element of the matrix. A

3. Matrices Each number in a matrix is called an element of the matrix. Matrix A has 8 elements arranged in 4 rows and 2 columns. Matrix B has 12 elements arranged in 4 rows and 3 columns. A matrix with m rows and n columns is called an m × n matrix . The expression m × n is called the size of the matrix and the numbers m and n are called the dimensions of the matrix. When giving the size of a matrix, the number of rows is given first.

4. Matrices A matrix having n rows and n columns is called a square matrix of order n . A matrix having only 1 column is called a column matrix . A matrix having only 1 row is called a row matrix .

5. Element Position Notation in Matrices The position of an element in a matrix is given by the row and column containing the element. This is usually denoted using double subscript notation a ij , where i is the row and j is the column containing the element a ij . Note that a 12 is read “ a sub one two” not “ a sub twelve.” The collection of elements a ii make up the principal diagonal of a matrix. In matrix A , a 11 = 1, and a 22 = 7 form the principal diagonal.

6. Matrices and Graphing Calculators Most graphing calculators are capable of storing and manipulating matrices. Matrix A as input into a graphing calculator is shown. Note that the highlighted entry is a 32 and the calculator uses a comma between the row and column location of the element.

7. Matrices and Systems of Equations Matrices serve as a shorthand for solving systems of linear equations. Consider the system of equations 3 x – 4 y = 7 2 x + 5 y = –2 There are three important matrices associated with this system. These matrices are the

8. Generalized Augmented Matrix for a System of Two Linear Equations in Two Variables Associated with the system of equations with variables x 1 and x 2 where x 1 and x 2 are variables, is the augmented matrix of the system Note, for example, that the entry a 21 is the coefficient for row 2 column 1.

9. Theorem 1 Operations That Produce Row-Equivalent Matrices An augmented matrix is transformed into a row-equivalent matrix by performing any of the following row operations : Two rows are interchanged ( R i ↔ R j ). A row is multiplied by a nonzero constant ( kR i → R i ). A constant multiple of one row is added to another row ( kR j + R i → R ik ). Note: The arrow → means “replaces.”

10. Example 1 Solve the System Using Augmented Matrix Methods Solution Write the augmented matrix corresponding to the system. The goal is to use row operations to transform the augmented matrix into the form where m and n are real numbers and the solution to the system is obvious.

11. Example 1 Solve the System Using Augmented Matrix Methods continued Step 1 To get a 1 in the upper left column, interchange R 1 and R 2 . ↔ Step 2 To get a 0 in the lower left corner, multiply R 1 by –3 and add to R 2 . ↔

12. Example 1 Solve the System Using Augmented Matrix Methods continued Step 3 To get a 1 in the second row, second column, multiply R 2 by 1/10. ↔ Step 4 To get a 0 in the first row second column, multiply R 2 by –2 and add to R 1 . ↔

13. Example 1 Solve the System Using Augmented Matrix Methods continued This last matrix in the process is the augmented matrix for the system The steps in the process result in a system that is equivalent to the original system and the solution is x 1 = 3 and x 2 = -2.

14. Example 1 Solve the System Using Augmented Matrix Methods continued We verify this result by checking the values in the original system. With x 1 = 3 and x 2 = –2, 3·(3) + 4·(–2) = 9 – 8 = 1 which checks in the first equation. In the second equation, 3 – 2·(–2) = 3 – (–4) = 3 + 4 = 7, which checks in the second equation.

15. Example 1A Solve the System Using Calculator Row Operations Solution Write the augmented matrix corresponding to the system and input the augmented matrix into the graphing calculator . As before, the goal is to use row operations to transform the augmented matrix into the form where m and n are real numbers and the solution to the system is obvious. We solve the same system using graphing calculator row operations.

16. Example 1A Solve the System Using Calculator Row Operations continued Step 1 To get a 1 in the upper left column, use the calculator rowSwap command to interchange R 1 and R 2 and store the result as [A]. Step 2 To get a 0 in the lower left corner, multiply R 1 by –2 and add to R 2 . Store the result at [A].

17. Example 1A Solve the System Using Calculator Row Operations continued Step 3 To get a 1 in the second row, second column, multiply R 2 by 1/10. Store the result as [A]. Step 4 To get a 0 in the first row second column, multiply R 2 by –2 and add to R 1 . Store the result as [A].

18. Example 1A Solve the System Using Calculator Row Operations continued Step 5 To get a 0 in the first row, second column, multiply R 2 by 2 and add the result to R 1 . Store the result as [A]. This last matrix corresponds to the system of equations The solution to this system is the the same as the original system.

19. Example 2 Solve the System Using Augmented Matrix Methods Solution Write the augmented matrix for the system. Step 1 To get a 1 in the upper left column, multiply R 1 by ½. Step 2 To get a 0 in the lower left corner, multiply R 1 by –3 and add to R 2 .

20. Example 1 Solve the System Using Augmented Matrix Methods continued Step 3 To get a 1 in the second row, second column, multiply R 2 by 2/17. Step 4 To get a 0 in the first row second column, multiply R 2 by 3/2 and add to R 1 . The solution to the system is x 1 = 3/2, x 2 = –1.

21. Example 3 Solve the System Using Augmented Matrix Methods Solution Write the augmented matrix for the system. Step 1 To get a 1 in the upper left column, multiply R 1 by 1/2 Step 2 To get a 0 in the lower left corner, multiply R 1 by 6 and add to R 2 .

22. Example 3 Solve the System Using Augmented Matrix Methods continued This matrix is the augmented matrix for the system This system is equivalent to the original system. The graphs of the two original equations coincide and there are infinitely many solutions. When an augmented matrix corresponding to a two equation system ends with a row of zeroes as a result of using the processes in Theorem 1, the system is dependent, and there are infinitely many solutions.

23. Example 3 Solve the System Using Augmented Matrix Methods continued We introduce a parameter to represent the infinitely many solutions. For any real number t , substituting t into the system gives a pair of values that solve the original system.

24. Example 4 Solve the System Using Augmented Matrix Methods Solution Write the augmented matrix for the system. Step 1 To get a 1 in the upper left column, interchange R 1 and R 2 . Step 2 To get a 0 in the lower left corner, multiply R 1 by –2 and add to R 2 .

25. Example 4 Solve the System Using Augmented Matrix Methods continued This last matrix is the augmented matrix for the system This system is equivalent to the original system. The second equation is not satisfied by any ordered pair of real numbers. The original system is inconsistent and has no solution. When an augmented matrix corresponding to a two equation system ends with all zeroes to the left of the vertical bar and a nonzero number to the right, the system is inconsistent and there are no solutions.

26. Summary Possible Final Matrix Forms for a System of Two Linear Equations in Two Variables m , n , and p are real numbers; p ≠ 0.