System of Linear Equations with Unique Solution Budi Murtiyasa Universitas Muhammadiyah Surakarta 1budi murtiyasa / linear equation.

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Presentation transcript:

System of Linear Equations with Unique Solution Budi Murtiyasa Universitas Muhammadiyah Surakarta 1budi murtiyasa / linear equation

2 System of linear Equations 2x 1 – x 2 + 2x 3 = 7 x 1 + 3x 2 – 5x 3 = 0 - x 1 + x 3 = 4 Using matrix = A X =G 3x 1 – 7x 2 + x 3 = 0 -2x 1 + 3x 2 – 4x 3 = 0 Using matrix = AX=G A, coeficient matrix X, variable matrix G, constant matrix

budi murtiyasa / linear equation3 SYSTEM OF LINEAR EQUATIONS A X = G G = 0 ? YES HOMOGENEOUS SYSTEM A X = 0 NO NONHOMOGENEOUS SYSTEM A X = G, where G ≠ 0 example : 3x – 5y + 3z = 0 x + 2y – z = 0 2x + y + 2z = 0 Example: 2x + y – 7z = 0 3x + 2y + z = 5 x – 6y + 2z = 0

budi murtiyasa / linear equation4 Nonhomogeneous SLE with unique (one) solution Find the solution of : x 1 – 2x 2 + x 3 = -5 3x 1 + x 2 – 2x 3 = 11 -2x 1 + x 2 + x 3 = -2 Using a inverse of matrix : 1. Find inverse of A (by adjoint matrix). The solution : Thus : x 1 = 2 x 2 = 3 x 3 = -1 A X = G A -1 A X = A -1 G X = A -1 G A =, then Adj(A) = det(A) = 6 A -1 =Adj(A)= 2. X = A -1 G X = =

budi murtiyasa / linear equation5 Solve the system using inverse matrix: x 1 – 2x 2 + x 3 = 0 -2x 1 + 3x 2 – 4x 3 = -8 5x 1 + x 2 – x 3 = -4

budi murtiyasa / linear equation6 Using inverse matrices, solve the system: x 1 – 2x 2 + x 3 = -5 3x 1 + x 2 – 2x 3 = 11 -2x 1 + x 2 + x 3 = -2

budi murtiyasa / linear equation7 Nonhomogenous SLE with unique solution Find a solution of : x 1 – 2x 2 + x 3 = -5 3x 1 + x 2 – 2x 3 = 11 -2x 1 + x 2 + x 3 = -2 Using Cramer Rule: 1.finding det(A), and det(A i ), that are the determinant of A by replacing the i th coloumn with constant matrix G The solution : |A| = = 6 | A 1 | = 2. X i = |A i | / | A | = 12 | A 2 | == 18 | A 3 | == - 6

budi murtiyasa / linear equation8 Solve the system : using cramer’s rule -5x 1 + 4x 2 – 2x 3 = -10 x 1 – 2x 2 + x 3 = 2 2x 1 + 3x 2 – 4x 3 = -8

budi murtiyasa / linear equation9 Solve the systems below using: (a)Inverse matrix (b)Cramer’s rule

budi murtiyasa / linear equation10 Solve the systems below using: (a)Inverse matrix (b)Cramer’s rule

budi murtiyasa / linear equation11 Solve the systems below using: (a)Inverse matrix (b)Cramer’s rule