8.1 Matrices and Systems of Equations

1. 8.1 Matrices and Systems of Equations

2. Let’s do another one: we’ll keep this one Now we’ll use the 2 equations we have with y and z to eliminate the y’s.

3. multiply the first equation by 8 and add the equations to eliminate y.

4. Matrix: rectangular organization of real numbers in rows and columns Column 1 C 2 C 3 C n Row 1 Row 2 Row 3 Row m • Entry aijis a real number in a certain “i” row and “j” column

5. Dimension, or ORDER, of a matrix • Dimension or order is given by m x n m = number of rows n = number of columns

6. Square Matrix • In a square matrix: m = n; rows = columns • Note: To solve a system of equations, you will need square coefficient matrices

7. Coefficient Matrix • A matrix whose real entries are the coefficients from a system of equations

8. Solution Matrix • A column matrix whose entries are the solutions of the system of equations

9. Augmented matrix- A matrix set up where the coefficient matrix and solution matrix of a system of equations is combinedNotation: Matrix square brackets with a bar separating the coefficients

10. Augmented Matrix cont.’d

11. Elementary Row Operations- Manipulating to Solve Matrices/ Systems 1.) Interchange two rows 2.) Multiply a row by a NONZERO multiple 3.) Add a multiplied row to another row

13. Row- Echelon Form (ref) Page 548: 1.) If there is a row of all zeros, it is the last row 2.) The first nonzero entry of a nonzero row is 1 3.) The 1’s occur at a diagonal Reduced Row-echelon Form (rref) If every column that has a leading 1 has zeros above and below the 1.

14. Page 548

15. Gaussian Elimination 1.) Write the augmented matrix of a system of equations 2.) Use elementary row operations to rewrite the augmented matrix in ref form 3.) Rewrite the matrix as a system of equations and back-substitute in to solve for each variable

16. Example 1: Solve Row 2: Added R1 + R2 -4 3 5 -1 3 -1 -3 0 -1 2 2 Row 3: Added -2(R1) + R3 -2 8 -6 -10 2 0 -4 6 0 8 -10 -4

17. Row 2: -1(R2) 0(-1) (-1)(-1) 2(-1) 2(-1) Row 3: Added 8(R2) + R3 0 -8 16 16 08 -10 -4 0 0 6 12 Row 3: 1/6(R3)

18. No put back into a system of equations and back substitute

19. 8.1 Pg. 554 #3-11(ODD); 13; 16; 19; 31-34; 58-62