1. 12-2: Matrices

2. 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 = y – 5z = 6 3x + 3y + 10z = -2
Can be written as
Note the 0 as the x-coefficient in the 2nd equation

3. 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

4. 12-2: Matrices -2r1 + r2 = 0 2 | 6 replace row 2 ½ r2 replace row 2
This represents the solution
x = -8, y = 3

5. 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

6. 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 = x + 0y = 0 Dependent system
1x + 0y + 0z = x + 1y + 0z = x + 0y + 1z = 4 (3, -7, 4) Independent system
1x + 2y = x + 0y = 3 Inconsistent system

7. 12-2: Matrices Assignment Page 801 – 802
Problems 1 – – 17 (show work) all odd problems

8. 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

9. 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

10. 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

11. 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. 12-2: Matrices Assignment Page 801 – 802
Problems 21 – all odd problems

13. 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 $ 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 = [Total cash]
0.18x y z = [Total interest]
y = 3x [3x as 15%]

14. 12-2: Matrices Get last equation into matrix form
x + y + z = [Total cash]
0.18x y z = [Total interest]
y = 3x [3x as 15%]
Get last equation into matrix form
-3x + y = 0
Turn equations into a matrix
Use calculator to calculate reduced row-echelon form
Borrowed 18% Borrowed 15% Borrowed 9%

15. 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