893 9.2 Matrix Solution of Linear Systems The Gauss-Jordan method is a systematic technique for applying matrix row transformations in an attempt to reduce a matrix to diagonal form, with 1s along the diagonal, from which the solutions are easily obtained. c 1 0 0 1 2 a bd or £ 1 0 0 0 1 0 0 0 1 3 a b c § Diagonal form, or reduced row-echelon form This form is also called reduced row-echelon form. Using the Gauss-Jordan Method to Transform a Matrix into Diagonal Form Step 1 Obtain 1 as the first element of the first column. Step 2 Use the first row to transform the remaining entries in the first column to 0. Step 3 Obtain 1 as the second entry in the second column. Step 4 Use the second row to transform the remaining entries in the second column to 0. Step 5 Continue in this manner as far as possible. NOTE The Gauss-Jordan method proceeds column by column, from left to right. In each column, we work to obtain 1 in the appropriate diagonal location, and then use it to transform the remaining elements in that column to 0s. When we are working with a particular column, no row operation should undo the form of a preceding column. EXAMPLE 1 Using the Gauss-Jordan Method Solve the system. 3x - 4y = 1 5x + 2y = 19 SOLUTION Both equations are in the same form, with variable terms in the same order on the left, and constant terms on the right. c 3 5 -4 2 2 1 19d Write the augmented matrix. The goal is to transform the augmented matrix into one in which the value of the variables will be easy to see. That is, because each of the first two columns in the matrix represents the coefficients of one variable, the augmented matrix should be transformed so that it is of the following form. c 1 0 0 1 ` k j d Here k and j are real numbers. In this form, the matrix can be rewritten as a linear system. x = k y = j This form is our goal.
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