Chapter 2 Part 2 MATRICES
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1 Finite Math B Chapter 2 MATRICES 1 Chapter 2 Part 2 MATRICES A: Augmented Matrices and Row Operations (Lessons 2.2 pg 68-70) Augmented Matrices Suppose you are given a system of equations such as: 2x y z 2 x 3y 2z 1 x y z 2 The system can be written as a matrix: To separate the coefficients of the variables from the constants after the equals signs, we draw in a vertical line in the matrix. This is called an augmented matrix. Example 1: Write an augmented matrix for each system of equations. Do not solve. a) 3x 4y 2 8x y4 b) 5x 25 x4y10 c) 2x 3y 5z 12 3x4z 15 x y z 5 Example 2: Write the system of equations associated with each augmented matrix. a) b)
2 Finite Math B Chapter 2 MATRICES 2 (example 2 continued. Write the system of equations associated with each augmented matrix) c) d) In algebra, our goal when faced with a system of equations is to find a solution for x, y, and z that makes the system of equations true. Note that when you see this pattern: # # # 3 You end up with x # 1 y # z # 2 3 (1 s on main diagonal, 0 s elsewhere, constants last column) Row Operations We will be using Row Operations to manipulate matrices to help us solve systems of equations. What you are allowed to do: 1. INTERCHANGE TWO ROWS 2. MULTIPLY THE ELEMENTS OF A ROW BY A NONZERO REAL NUMBER 3. ADD A NONZERO MULTIPLE OF THE ELEMENTS OF ONE ROW TO THE CORRESPONDING ELEMENTS OF A NONZERO MULTIPLE OF SOME OTHER ROW. Example 3: Use the indicated row operations to change each matrix. a) Interchange R 1 with R
3 Finite Math B Chapter 2 MATRICES 3 (Example 3 continued: Use the indicated row operation to change each matrix) b) Replace R 3 by 1 3 R c) Replace R 2 with (-2)R 1 +R d) Replace R 3 with (-3)R 2 + 5R e) Replace R 1 with (-2)R 3 + 3R f) Replace R 2 with (-7)R 3 + 6R g) Replace R 2 with (-1)R 1 +4R
4 Finite Math B Chapter 2 MATRICES 4 B: Gauss-Jordan Method for Solving Systems of Equations (Lesson 2.2, textbook pg 70 80) Problem: Find the solution to a system of equations like x y 5z 6 3x 3y z 10 x 3y 2z 5 Strategy: 1. Write the system of equations as an augmented matrix 2. Use Row Operations to transform the matrix into a matrix with whole numbers on the main diagonal, but 0 s elsewhere. 3. Use Row Operations to transform the matrix into an identity matrix. (1 s on diagonal, 0 s elsewhere) 4. Final solution = numbers in the answer column of the matrix. Example: Example: Meaning: Meaning: Make it happen: GAUSS-JORDAN Your Goal: 1 0 # 0 1 # or # # # Using legal row operations: 2x2 System Clear Col 1 Clear Col 2 Create 1 s Main Diag. # # # # 0 # 1 0 # 0 # # 0 # # 0 1 # 3x3 System Clear Col 1 Clear Col 2 Clear Col 3 Create 1 s Main Diag. # # # # # 0 # # # 0 0 # # 0 # # # 0 # # # 0 # 0 # # 0 # # # 0 0 # # 0 0 # # #
5 Finite Math B Chapter 2 MATRICES 5 Example 1 : Use the Gauss-Jordan Method to solve each system of equations a) 2x 4y 2 3x5y 0 b) 3x4y 1 5x2y 19 c) x2y 2 3x6y 5
6 Finite Math B Chapter 2 MATRICES 6 (Example 1 continued): Use the Gauss-Jordan Method to solve each system of equations d) x y z 3 2x 3y 7z 0 x 3y 2z 17
7 Finite Math B Chapter 2 MATRICES 7 (Example 1 Continued): Use the Gauss-Jordan Method to solve each system of equations e) 2x 5y 4z 8 2x2z 4 x 2y z 2
8 Finite Math B Chapter 2 MATRICES 8 (Example 1 Continued): Use the Gauss-Jordan Method to solve each system of equations f) x y 5z 6 3x 3y z 10 x 3y 2z 5
9 Finite Math B Chapter 2 MATRICES 9 C: Matrix Inverses (Lessons 2.5 pg ) Additive Inverse Vs Multiplicative Inverse For numbers, algebraic expressions, and matrices zero or the zero matrix plays the role of the Additive Identity. The sum of a value and the additive identity is the original value, unchanged x If A and B are additive inverses then their sum gives you the additive inverse value, then A + B = 0 Examples of Additive Inverses: = 0-3x + = = For numbers and algebraic expressions, 1 plays the role of the Multiplicative Identity. For matrices, the Identity Matrix is the Multiplicative Identity. The product of the a value and the multiplicative identity is the original value, unchanged. 144 x 1 = 3 x 1 = If A and B are Multiplicative Inverses then their product gives you the multiplicative inverse value, then A x B = 1 Examples of Multiplicative Inverses: 144 x = 1 3 x = ???? = Finding a multiplicative inverse for a matrix is more challenging than just reciprocating each value in the matrix due to the unique ROWS x COLUMNS requirement of multiplying matrices.
10 Finite Math B Chapter 2 MATRICES 10 Terminology/Facts Only a square matrix can have a multiplicative inverse. 1 If A is your given matrix, then A is the multiplicative matrix. I is the identity matrix. The identity matrix I is a square matrix with 1 s on the main diagonal and 0 s elsewhere 1 AA I AI A Example 1: Decide whether the given matrices are inverses of each other. (Check to see if their product is the identity matrix I ) a) and b) and c) and Finding the Multiplicative Inverse of a Matrix Step 1: Form an augmented matrix that looks like AI Step 2: Perform row operations to get the matrix in the form IBif possible. Step 3: Matrix B is A 1
11 Finite Math B Chapter 2 MATRICES 11 Example 2: Find the multiplicative inverse of each matrix a) b) c)
12 Finite Math B Chapter 2 MATRICES 12 (Example 2 continued) : Find the multiplicative inverse of each matrix d) e)
13 Finite Math B Chapter 2 MATRICES 13 D: Using Matrix Inverses to Solve Systems of Equations (Lessons 2.5 pg ) Any system of equations can be written as AX where A the square coefficient matrix X the column matrix of the variables B the matrix of constants B Example: 2x5y15 x4y9 Solving A System of Equations Using the Inverse Method 1 1. Find the A, the inverse of the coefficient matrix 1 2. Multiply A B 3. Check your answers Example 1: Solve each system of equations by using the inverse of the coefficient matrix. a) 2x5y15 x4y9
14 Finite Math B Chapter 2 MATRICES 14 (Example 1 Continued): Solve each system of equations by using the inverse of the coefficient matrix. b) 2x7y14 3x4y8 c) x y z 1 4x 5y 2 y3z 3 We already found the inverse of this matrix in exercise 2d on page 12. Go back and copy that answer here:
15 Finite Math B Chapter 2 MATRICES 15 (Example 1 continued): Solve each system of equations by using the inverse of the coefficient matrix. d) x 2z 1 yz 5 x y 8
16 Finite Math B Chapter 2 MATRICES 16
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