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

A SYSTEMS WITH UNIQUE SOLUTION Consider the system: Augmented matrix: The system has unique solution:

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,

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

SOLVING SYSTEMS OF EQUATIONS Systems of linear equations :

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

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

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:

THE INVERSE OF THE COEFFICIENT MATRIX Solution:

THE INVERSE OF THE COEFFICIENT MATRIX Find A-1 : Cofactor of A : Therefore:

THE INVERSE OF THE COEFFICIENT MATRIX Find X :

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

GAUSS ELIMINATION Consider the systems of linear eq:

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

GAUSS ELIMINATION Example: Solve the system by using Gauss Elimination method:

GAUSS ELIMINATION Solution: Write in augmented form:

Reduce to REF : (Diagonal = 1)

x y z

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)

Reduce to RREF : (Diagonal = 1, Other entries = 0)

Example 2: Solve the system by Gauss elimination. Answer :

Gauss jordan elimination Example 4: Solve the system by Gauss Jordan elimination. Answer :

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

CRAMER’S RULE If : Then :

CRAMER’S RULE

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

CRAMER’S RULE Solution Determinant of A :

CRAMER’S RULE

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