Objectives  Represent systems of equations with matrices  Find dimensions of matrices  Identify square matrices  Identify an identity matrix  Form an augmented matrix  Identify a coefficient matrix  Reduce a matrix with row operations  Reduce a matrix to its row-echelon form  Solve systems of equations using the Gauss-Jordan elimination method

Matrix Representation of Systems of Equations When given a system of equations, it can be written as a matrix. The column to the right of the vertical line, containing the constants of the equations, is called the augment of the matrix, and a matrix containing an augment is called an augmented matrix.

Any augmented matrix that has 1’s or 0’s on the diagonal of its coefficient part and 0's below the diagonal is said to be in row-echelon form.

Example Solve the system Solution Begin by writing the augmented matrix.

Example (cont) Interchange equations 1 and 2; thus change rows 1 and 2. Get 0 as the first entry in the second row and the first entry of the third row. –2R 1 + R 2 →R 2 –3R 1 + R 3 →R 3

Example (cont) –1R 2 → R 2 –4R 2 + R 3 → R 3 (–1/19)R 3 → R 3

Example (cont) The matrix is now in row-echlon form. The equivalent system can be solved by back substitution. The solution is (2, 1,  2).

Gauss-Jordan Elimination The augmented matrix representing n equations in n variables is said to be in reduced row-echelon form if it has 1’s or 0’s on the diagonal of its coefficient part and 0’s everywhere else.

Example Solve the system Solution Represent by the augmented matrix.

Example (cont) We can enter this augmented matrix into a graphing calculator and reduce the matrix to row-echelon form. x = 1, y = 11, z = –4, w = –5, or (1, 11, –4, –5)

Dependent and Inconsistent Systems A system with fewer equations than variables has either infinitely many solutions or no solutions. If a row of the reduced row-echelon coefficient matrix associated with a system contains all 0’s and the augment of that row contains a nonzero number, the system has no solution and is an inconsistent system. If a row of the reduced 3 × 3 row-echelon coefficient matrix associated with a system contains all 0’s and the augment of that row also contains 0, then there are infinitely many solutions and is a dependent system.

Example Ace Trucking Company has an order for delivery of three products: A, B, and C. If the company can carry 30,000 cubic feet and 62,000 pounds and is insured for $276,000, how many units of each product can be carried?

Example (cont) If we represent the number of units of product A by x, the number of units of product B by y, and the number of units of product C by z, then we can write a system of equations to represent the problem.

Example (cont) The Gauss-Jordan elimination method gives

Example (cont) To save time, if we use a graphing calculator. The solution to this system is x = –560 + z, y = 2000 – 2.5z, with the values of z limited so that all values are nonnegative integers. Product C: 560 ≤ z ≤ 800 (z is an even integer) Product B: y = 2000 – 2.5z Product A: x = –560 + z

Assignment Pg #15-21 (Must show work) #23-31 (May use the calculator) #34, #39 and #42