1. Matrices and Systems of Linear Equations
Section 7.1
Matrices and Systems of Linear Equations

2. Matrices
A matrix is a rectangular array of numbers written within brackets
Open books to page 562

3. Order of a Matrix Order of a Matrix: Row x Column 3 rows, 2 columns
1 row, 3 columns
1 x 3

4. Augmented Matrices
Matrices can be used as shorthand for systems of equations. When done so, they are called augmented matrices.
Each row is an equation
Vertical line represents the equal sign
First column is coefficients on the x
Second column is coefficients on the y
Constants to the right of the vertical line
Any variable not in the equation has an implied coefficient of 0

5. Write the system as an augmented matrix

6. Row Operations (Solving Systems)
Interchange any 2 rows
Multiply a row by a nonzero constant
Add a multiple of 1 row to another

7. Perform the row operation

8. Perform the row operation

9. Perform the row operation

10. Perform the row operation

11. Row-Echelon Form of a Matrix
Rows consisting entirely of 0’s are at the bottom of the matrix
For each row that does not consist entirely of 0’s, the first (leftmost) nonzero entry is 1 (called the leading 1)
The leading 1 in each row must have all zeros underneath it.

12. Determine whether the matrices are in Row-Echelon Form
Yes No
Yes No

13. Rewrite the Matrix in Row Echelon Form

14. Solve the system using Gaussian Elimination
Step 1: Write as an augmented matrix
Step 2: Use row operations to write in row-echelon form.
Need a 0 below the leading 1 in row 1

15. …continued Need a leading 1 in row 2 (turn the -5 into a 1)
Step 3: Write the augmented matrix as a system of equations.
Step 4: Back substitute to find all other variables.

16. Solve the system
and

18. Infinitely Many and No Solutions
Row 3 equation would say:
0x + 0y + 0z = 0
0 = 0
Infinitely Many Solutions on a line
Row 3 equation would say:
0x + 0y + 0z = 4
0 = 4
No Solutions