System of Linear Equations with Unique Solution Budi

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

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

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

Nonhomogeneous SLE with unique (one) solution Find the solution of : x 1 – 2 x 2 + x 3 = -5 3 x 1 + x 2 – 2 x 3 = 11 -2 x 1 + x 2 + x 3 = -2 det(A) = 6 A-1 = Adj(A) = Using a inverse of matrix : AX=G A-1 A X = A-1 G 2. X = A-1 G X= 1. Find inverse of A (by adjoint matrix). A= , then Adj(A) = budi murtiyasa / linear equation = Thus : x 1 = 2 x 2 = 3 x 3 = -1 The solution : 4

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

Using inverse matrices, solve the system: x 1 – 2 x 2 + x 3 = -5 3 x 1 + x 2 – 2 x 3 = 11 -2 x 1 + x 2 + x 3 = -2 budi murtiyasa / linear equation 6

Nonhomogenous SLE with unique solution Find a solution of : | A 2 | = x 1 – 2 x 2 + x 3 = -5 3 x 1 + x 2 – 2 x 3 = 11 -2 x 1 + x 2 + x 3 = -2 | A 3 | = Using Cramer Rule: 1. finding det(A), and det(Ai), that are the determinant of A by replacing the ith coloumn with constant matrix G |A| = | A 1 | = = 18 =-6 2. Xi = |Ai | / | A | =6 = 12 The solution : budi murtiyasa / linear equation 7

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

Solve the systems below using: (a)Inverse matrix (b)Cramer’s rule budi murtiyasa / linear equation 9

Solve the systems below using: (a)Inverse matrix (b)Cramer’s rule budi murtiyasa / linear equation 10

Solve the systems below using: (a)Inverse matrix (b)Cramer’s rule budi murtiyasa / linear equation 11