Section 3.7 Solving Systems of Linear Equations by Using Matrices 295 2. Solving Systems of Linear Equations by Using Elementary Row Operations The following elementary row operations performed on an augmented matrix produce an equivalent augmented matrix: • Interchange two rows. • Multiply every element in a row by a nonzero real number. • Add a multiple of one row to another row. When we are solving a system of linear equations by any method, the goal is to write a series of simpler but equivalent systems of equations until the solution is obvious. The Gauss-Jordan method uses a series of elementary row operations performed on the augmented matrix to produce a simpler augmented matrix. In particular, we want to produce an augmented matrix that has 1’s along the diagonal of the matrix of coefficients and 0’s for the remaining entries in the matrix of coefficients.A matrix written in this way is said to be written in reduced row echelon form. For example, the augmented matrix from Example 3(c) is written in reduced row echelon form. 4 1 0 The solution to the corresponding system of equations is easily recognized as Similarly, matrix B represents a solution of and . B c1 0 0 1 x a y b a b ` d x 4, y 1, and z 0. £ 1 0 0 0 1 0 0 0 1 † § the Gauss-Jordan Method We know that interchanging two equations results in an equivalent system of linear equations. Interchanging two rows in an augmented matrix results in an equivalent augmented matrix. Similarly, because each row in an augmented matrix represents a linear equation, we can perform the following elementary row operations that result in an equivalent augmented matrix.
miller_intermediate_algebra_4e_ch1_3
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