SECTION 11.2 Systems of Linear Equations: Matrices 771 Row Operations • Interchange any two rows. • Replace a row by a nonzero multiple of that row. • Replace a row by the sum of that row and a constant nonzero multiple of some other row. If we do not include the constants to the right of the equal sign (that is, to the right of the vertical bar in the augmented matrix of a system of equations), the resulting matrix is called the coefficient matrix of the system. For the systems discussed in Example 1, the coefficient matrices are − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ − ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ 3 2 4 3 and 2 1 1 1 0 2 1 1 0 Now Work PROBLEM 9 Writing the System of Linear Equations from the Augmented Matrix Write the system of linear equations that corresponds to each augmented matrix. (a) − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 5 3 2 1 13 10 (b) ⎡ − − ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ 3 2 0 1 0 1 1 2 1 7 8 0 EXAMPLE 2 Solution (a) The augmented matrix has two rows and so represents a system of two equations. The two columns to the left of the vertical bar indicate that the system has two variables. If x and y are used to denote these variables, the system of equations is (1) (2) + = − + =− ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 5 2 13 3 10 (b) Since the augmented matrix has three rows, it represents a system of three equations. Since there are three columns to the left of the vertical bar, the system contains three variables. If x y , , and z are the three variables, the system of equations is − − = + = + = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ (1) (2) (3) x y z x z y z 3 7 2 2 8 0 2 Write the System of Equations from the Augmented Matrix 3 Perform Row Operations on a Matrix Row operations on a matrix are used to solve systems of equations when the system is written as an augmented matrix. There are three basic row operations. These three row operations correspond to the three rules given earlier for obtaining an equivalent system of equations. When a row operation is performed on a matrix, the resulting matrix represents a system of equations equivalent to the system represented by the original matrix. For example, consider the augmented matrix − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 1 4 2 1 3 2
RkJQdWJsaXNoZXIy NjM5ODQ=