772 CHAPTER 11 Systems of Equations and Inequalities Suppose we want to use a row operation that results in a matrix whose entry in row 2, column 1 is 0. The row operation to use is − Multiply each entry in row 1 by 4,and add the result to the corresponding entries in row 2. (2) If we use R2 to represent the new entries in row 2 and r1 and r2 to represent the original entries in rows 1 and 2, respectively, we can represent the row operation in statement (2) by = − + R r r 4 2 1 2 Then ( ) − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ → −⋅+ −⋅+− −⋅+ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 1 4 2 1 3 2 1 2 4 1 4 4 2 1 3 4 3 2 1 0 2 9 3 10 ↑ =− + R r r 4 2 1 2 We now have the entry 0 in row 2, column 1. Row 2 represents the equation − = − y9 10 since the coefficient of x is now zero. Now Work PROBLEM 19 Using a Row Operation on an Augmented Matrix Use the row operation = − + R r r 3 2 1 2 on the augmented matrix − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 1 3 2 5 2 9 EXAMPLE 3 Solution The row operation = − + R r r 3 2 1 2 replaces the entries in row 2 by the entries obtained after multiplying each entry in row 1 by −3 and adding the result to the corresponding entries in row 2. ( ) ( ) − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ → − −⋅+ −⋅− +− −⋅+ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 1 3 2 5 2 9 1 2 3 1 3 3 2 5 2 3 2 9 1 0 2 1 2 3 ↑ =− + R r r 3 2 1 2 Row 2 represents the equation = y 3 since the coefficient of x is now zero. Finding a Row Operation Find a row operation that results in the augmented matrix ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 1 0 2 1 2 3 having 0 in row 1, column 2. EXAMPLE 4 Solution We want 0 in row 1, column 2. Because the entry in row 2, column 2 is 1, multiply row 2 by 2 and add the result to row 1. That is, use the row operation = + R r r 2 . 1 2 1 ( ) ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ → ⋅ + ⋅ + − ⋅ + ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 1 0 2 1 2 3 2 0 1 2 1 2 0 1 2 3 2 3 1 0 0 1 8 3 ↑ = + R r r 2 1 2 1 A word about notation:The row operation = + R r r 2 1 2 1 changes the entries in row 1.We change the entries in row 1 by multiplying the entries in some other row by a nonzero number and adding the results to the original entries of row 1.
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