管理数学ppt.pptVIP

x1 x2 x3 X = 1 3 -2 B = 1 -1 -2 2 -3 -5 -1 3 5 A = Matrix Representation AX = B 3x1 + 2x2 - 5x3 = 7 x1 - 8x2 + 4x3 = 9 2x1 + 6x2 - 7x3 = -2 Transfer (I) to matrix format. (I) 3 2 -5 1 -8 4 2 6 -7 A= x1 x2 x3 X= 7 9 -2 B= Example 2 Matrix of Coefficients 1 1 1 2 3 1 1 -1 -2 x1+x2+x3=2 2x1+3x2+x3=3 x1-x2-2x3=-6 Coefficients of the system or Matrix A Augmented Matrix x1+x2+x3=2 2x1+3x2+x3=3 x1-x2-2x3=-6 1 1 1 2 2 3 1 3 1 -1 -2 -6 Coefficients and RHS, or [ A | B ]. Reduced Echelon Form 1. Any rows with all zeros are at the bottom. 2. Leading 1. 3. Leading 1 to the right. 4. All other elements in a leading 1 column are zeros. 1 0 8 0 1 2 0 0 0 1 2 0 4 0 0 0 0 0 0 1 3 1 0 0 2 0 0 1 4 0 1 0 3 1 2 3 0 0 0 0 1 0 0 0 0 Examples I 1 2 0 3 0 0 0 3 4 0 0 0 0 0 1 1 7 0 8 0 1 0 3 0 0 1 2 0 0 0 0 Examples II Elementary Row Operations 1. Interchange two rows 2. Multiply the elements of a row by a nonzero constant 3. Add a multiple of the elements of one row to the corresponding elements of another row. x1 + x2 + x3 = 2 --------(1) 2x1 + 3x2 + x3 = 3 ------(2) x1 - x2 - 2x3 = -6 --------(3) 1 1 1 2 2 3 1 3 1 -1 -2 -6 Continuous Operations 1 1 1 2 2 3 1 3 1 -1 -2 -6 Interchange r.1 and r.3. Multiply r.2 by 1/2. Add a -1 multiple r.2 to r.3. 1 -1 -2 -6 1 1.5 0.5 1.5 0 -0.5 0.5 0.5 Result 1 1 1 2 2 3 1 3 1 -1 -2 -6 Through Elementary Row operations The Objective Reduced echelon Form ? ? ? Equivalent Systems Suppose that A and B are both systems of linear equations. A and B are equivalent if they are related through elementary transformations. A and B has the same solution if they are equivalent. Solving a system of linear equations Gauss-Jordan Elimination Gauss Elimination Gauss-Jordan Elimination 1. Write the augmented matrix. 2. Derive the