TYPES OF SOLUTIONS SOLVING EQUATIONS

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Presentation transcript:

TYPES OF SOLUTIONS SOLVING EQUATIONS CHAPTER 2 MATRICES TYPES OF SOLUTIONS SOLVING EQUATIONS

TYPES OF SOLUTIONS TO SYSTEMS OF LINEAR EQUATIONS There are 3 possible solutions: 3 TYPES OF SOLUTIONS A SYSTEM WITH UNIQUE SOLUTION A SYSTEM WITH INFINITELY MANY SOLUTIONS A SYSTEM WITH NO SOLUTION NHAA/IMK/UNIMAP

A SYSTEMS WITH UNIQUE SOLUTION Consider the system: Augmented matrix: The system has unique solution: NHAA/IMK/UNIMAP

A SYSTEMS WITH INFINITELY MANY SOLUTION Consider the system: Augmented matrix: The system has many solutions: let where s is called a free variable. Then, NHAA/IMK/UNIMAP

A SYSTEMS WITH NO SOLUTION Consider the system: Augmented matrix: The system has no solution, since coefficient of is ‘0’. NHAA/IMK/UNIMAP

SOLVING SYSTEMS OF EQUATIONS Systems of linear equations : NHAA/IMK/UNIMAP

SOLVING SYSTEMS OF EQUATIONS 4 methods used to solve systems of equations. The Inverse of the Coefficient Matrix Gauss Elimination Gauss-Jordan Elimination Cramer’s Rule NHAA/IMK/UNIMAP

SOLVING SYSTEMS OF EQUATIONS Matrix Form: AX = B To find X: X =A-1 B NHAA/IMK/UNIMAP

THE INVERSE OF THE COEFFICIENT MATRIX Method : X =A-1 B Example: Solve the system by using A-1 , the inverse of the coefficient matrix: NHAA/IMK/UNIMAP

THE INVERSE OF THE COEFFICIENT MATRIX Solution: NHAA/IMK/UNIMAP

THE INVERSE OF THE COEFFICIENT MATRIX Find A-1 : Cofactor of A : Therefore: NHAA/IMK/UNIMAP

THE INVERSE OF THE COEFFICIENT MATRIX Find X : NHAA/IMK/UNIMAP

THE INVERSE OF THE COEFFICIENT MATRIX Example 2: Solve the system by using A-1 , the inverse of the coefficient matrix: Answer : NHAA/IMK/UNIMAP

GAUSS ELIMINATION Consider the systems of linear eq: NHAA/IMK/UNIMAP

GAUSS ELIMINATION Write in augmented form : [A|B] Using ERO, such that A may be reduce in REF/Upper Triangular NHAA/IMK/UNIMAP

GAUSS ELIMINATION Example: Solve the system by using Gauss Elimination method: NHAA/IMK/UNIMAP

GAUSS ELIMINATION Solution: Write in augmented form: NHAA/IMK/UNIMAP

Reduce to REF : (Diagonal = 1) NHAA/IMK/UNIMAP

x y z NHAA/IMK/UNIMAP

GAUSS JORDAN ELIMINATION Written in augmented form : [A|B] Using ERO, such that A may be reduce in RREF/IDENTITY (DIAGONAL = 1, OTHER ENTRIES = 0) NHAA/IMK/UNIMAP

Reduce to RREF : (Diagonal = 1, Other entries = 0) NHAA/IMK/UNIMAP

NHAA/IMK/UNIMAP

Example 2: Solve the system by Gauss elimination. Answer : NHAA/IMK/UNIMAP

Gauss jordan elimination Example 4: Solve the system by Gauss Jordan elimination. Answer : NHAA/IMK/UNIMAP

CRAMER’S RULE Theorem 5 Cramer’s Rule for 3x3 system Given the system: with : NHAA/IMK/UNIMAP

CRAMER’S RULE If : Then : NHAA/IMK/UNIMAP

CRAMER’S RULE NHAA/IMK/UNIMAP

CRAMER’S RULE Example 5: Solve the system by using the Cramer’s Rule. NHAA/IMK/UNIMAP

CRAMER’S RULE Solution Determinant of A : NHAA/IMK/UNIMAP

CRAMER’S RULE NHAA/IMK/UNIMAP

CRAMER’S RULE Example 6: Solve the system by using Cramer’s Rule. Answer : NHAA/IMK/UNIMAP