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Published byColleen Wilkerson Modified over 7 years ago

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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

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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.

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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.

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

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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

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

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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).

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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.

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Example Solve the system Solution Represent by the augmented matrix.

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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)

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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.

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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?

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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.

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Example (cont) The Gauss-Jordan elimination method gives

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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

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Assignment Pg. 518-521 #15-21 (Must show work) #23-31 (May use the calculator) #34, #39 and #42

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