1. Lesson 13-1: Matrices & Systems Objective: Students will: State the dimensions of a matrix Solve systems using matrices

2. Matrix An array of numbers aligned in rows & columns (rows are always first) a 11 = number in 1 st row, 1 st column Dimensions Stated as “rows Χ columns” (rows are always first) Dimensions:3 X 5

3. Using Matrices to Solve Systems ► Use Standard Form: Ax + By = C ► Put coefficients and constant in a matrix ► Solve by getting zeros in the lower left diagonal - 0’s below the main diagonal is called triangular form ► Allowable operations - interchanging 2 rows - multiplying a row by a constant (not zero) - adding 2 rows replacing a row with the result ► Write equations in x-y form from triangular form ► Solve – substitute - solve

4. Example 1 Solve x + 2y = 3 3x + 8y = 1 ●-3add & replace 2 nd row Triangular form Main diagonal x’sy’s 2y = -8 y = -4 Now use this -3x -6(-4) = -9 -3x +24 =-9 x = 11 Now we are done!!! Since we are down to 1 variable and answer we can convert back and solve

5. Example 2 Solve2x + 4y + 8z = 6 x + 3y + 5z = 4 3x + 8y + 6z = 19 Then add 2 and 3 replace 3 ●-3 ●-2 Then add 1and 2 replace 2 ●3 ●-6 Then add 2 and 3 replace 3 Since we are down to 1 variable and answer we can convert back and solve xyz

6. Cont….. 48z = -48 z = -1 Now use this -6y -6(-1) = -6 -6y = -12 y = 2 Now use this 6x +12(2)+24(-1)= 18 6x =18 x = 3 Now we are done!!! Remember- You can put these back into the original 3 equations to make sure they are the solution to all of them. I know I did!!!

7. You try: 5x - 2y = -44 x + 5y = 2

8. 3X3 x - 2y + 3z = 4 2x - y + z = -1 4x + y + z = 1

9. Assignment 13-1/572/1-13 odd