Numerical Solutions of Linear Systems of Equations

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EE 216 Class NotesFrom Row 3, 176z = -704Therefore, z = - 704/176 = - 4From Row 2, 19y – 2z = 103or, 19y – 2 (- 4) = 103or, y = 5From Row 1, - 4x + 3y + 2z = -5or, - 4x + 3 (5) + 2 (- 4) = -5or, x = 3Elementary operations for equations and matrices are:(I)(II)(III)Interchange of two equations.Multiplication of an equation by a nonzero constant.Addition of a constant multiple of one equation to another equation.When would you interchange two equations (rows)?Let us consider the following set of equations.3y + 2z = 16 … … (1)4x + 2y - 3z = - 10 … … (2)3x + 4y + z = 9 … … (3)The corresponding augmented matrix is:⎛04⎜⎜⎜⎝33242− 3116⎞− 109⎟⎟⎟⎠Pages 6 of 21
EE 216 Class NotesEqn. (1) (Row 1) cannot be used to eliminate x from Eqns. (2) and (3) (Rows 2 and 3).Interchange Row 1 with Row 2.The augmented matrix becomes:⎛40⎜⎜⎜⎝3234− 321− 10⎞169⎟⎟⎟⎠Now follow the steps mentioned earlier to solve for the unknowns.The solution is:x = - 3y = 4z = 2Let us consider the following set of linear equations.2x 1 – 4x 2 + 5x 3 = 36 … … (1)- 3x 1 + 5x 2 + 7x 3 = 7 … … (2)5x 1 + 3x 2 – 8x 3 = - 31 … … (3)Gauss elimination suggests doing elementary row operations from top to bottom. Aslightly modified form, known as Gauss-Jordan elimination suggests doing elementaryrow operations from bottom to top as well.Pages 7 of 21
Page 3: EE 216 Class NotesGAUSS ELIMINATION
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Page 15 and 16: EE 216 Class NotesBased on the abov
Page 17 and 18: EE 216 Class Notes[A][C] = [B]where
Page 19 and 20: EE 216 Class NotesIt should be note
Page 21: EE 216 Class NotesSolve the followi

Numerical Solutions of Linear Systems of Equations

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