12-2: Matrices

12-2: Matrices Augmented matrix: each row of the matrix represents an equation of the system Numbers in 1st column are coefficients of x Numbers in 2nd column are coefficients of y etc… Last column represents the constant terms Ex 1: Writing a System as an Augmented Matrix x + 2y + 3z = -2 2y – 5z = 6 3x + 3y + 10z = -2 Can be written as Note the 0 as the x-coefficient in the 2nd equation

12-2: Matrices Solving Systems Using Augmented Matrices Elementary Row Operations You can: Interchange any two rows Replace any row with a nonzero constant multiple of itself Replace any row with the sum of itself and a nonzero constant multiple of another row Ex 2: Solve the system of equations 2x + 2y = -2 2x + 6y = -2 Write it as a matrix: Operations next slide

12-2: Matrices -2r1 + r2 = 0 2 | 6 replace row 2 ½ r2 replace row 2 This represents the solution x = -8, y = 3

12-2: Matrices The matrix is an example of a reduced row-echelon matrix. The conditions of a reduced row-echelon matrix are: All rows consisting entirely of zeros (if any) are at the bottom. The first nonzero entry in each nonzero row is a 1 (called a leading 1) Any column containing a leading 1 has zeros as all other entries Each leading 1 appears to the right of leading 1’s in any preceding row

12-2: Matrices Example 3: Using Gauss-Jordan Elimination The matrices below are in reduced row-echelon form. Write the system represented by each matrix, find the solutions, and classify the solution as independent, dependent, or inconsistent 1x – 3y = 4 0x + 0y = 0 Dependent system 1x + 0y + 0z = 3 0x + 1y + 0z = -7 0x + 0y + 1z = 4 (3, -7, 4) Independent system 1x + 2y = -1 0x + 0y = 3 Inconsistent system

12-2: Matrices Assignment Page 801 – 802 Problems 1 – 9 13 – 17 (show work) all odd problems

12-2: Matrices Entering a matrix in a graphing calculator (Casio) F1 [Mat] Pick a letter/name for the matrix you’re entering Choose your dimensions (rows x columns) Enter in your matrix values Exit out to the main screen To put a matrix in reduced row-echelon form OPTN F2 [MAT], F6 [MORE], F5 [Rref] F6 [MORE], F1 [MAT] (the name of your matrix) EXE

12-2: Matrices Entering a matrix in a graphing calculator (TI-86) 2nd, 7 [Matrix] F2 [edit] Pick a letter/name for the matrix you’re entering Choose your dimensions (rows x columns) Enter in your matrix values Exit out to the main screen To put a matrix in reduced row-echelon form F4 [OPS] F4 [rref] (the name of your matrix) ENTER

12-2: Matrices Ex 4: Solve the following systems using a calculator’s reduced row-echelon form feature. x = -3, y = 6 x = 4, y = 0, z = -3

12-2: Matrices Calculator solution to an inconsistent system Solving the system of equations Gets the solution The bottom row represents the equation 0x + 0y + 0z = 1, which has no solution and is therefore the system is inconsistent

12-2: Matrices Assignment Page 801 – 802 Problems 21 – 33 all odd problems

12-2: Matrices Ex 7: Applications using Rref Charlie is starting a small business and borrows $10,000 on three different credit cards, with annual interests of 18%, 15%, and 9%, respectively. He borrows three times as much on the 15% card as he does on the 18% card, and his total annual interest on all three cards is $1244.25. How much did he borrow on each credit card? Let x = amount borrowed at 18% Let y = amount borrowed at 15% Let z = amount borrowed at 9% Three equations x + y + z = 10000 [Total cash] 0.18x + 0.15y + 0.09z = 1244.25 [Total interest] y = 3x [3x as much @ 15%]

12-2: Matrices Get last equation into matrix form x + y + z = 10000 [Total cash] 0.18x + 0.15y + 0.09z = 1244.25 [Total interest] y = 3x [3x as much @ 15%] Get last equation into matrix form -3x + y = 0 Turn equations into a matrix Use calculator to calculate reduced row-echelon form Borrowed $1275 @ 18% Borrowed $3825 @ 15% Borrowed $4900 @ 9%

12-2: Matrices Assignment Page 801 – 802 Problems 37 – 45 (all) Skip #43 You must show the equations you used (or matrix you’re calculating the reduced row- echelon form) for credit