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Published byPearl Hubbard 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 ______________________of the matrix, and a matrix containing an augment is called an ______________________.

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A square matrix that has 1’s down its diagonal and 0’s everywhere else, like matrix I below, is called an _________________.

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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)

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______________ 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.

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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 ________________. 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 _________________.

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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.

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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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