English | Español

# Try our Free Online Math Solver! Online Math Solver

 Depdendent Variable

 Number of equations to solve: 23456789
 Equ. #1:
 Equ. #2:

 Equ. #3:

 Equ. #4:

 Equ. #5:

 Equ. #6:

 Equ. #7:

 Equ. #8:

 Equ. #9:

 Solve for:

 Dependent Variable

 Number of inequalities to solve: 23456789
 Ineq. #1:
 Ineq. #2:

 Ineq. #3:

 Ineq. #4:

 Ineq. #5:

 Ineq. #6:

 Ineq. #7:

 Ineq. #8:

 Ineq. #9:

 Solve for:

 Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:

# Systems of Linear Equations #2: Gaussian Elimination

Outline: Gaussian Elimination (Ax=b → Ux=c)
1) Review mechanics
2) Review modes of failures
3) The general pattern of GE
3) GE using Matrices (E and P)
4) General Rules of Matrix -Matrix Operations
Matrix Multiplication : AB
Matrix Powers : AP

Gaussian Elimination on a 3x3 system of equations :
Example ( with row exchange):
x + 2y + 4z = 1
2x + 4y + 2z = 2
6x + 10y - z = 8

Matrix Vector form:

Augmented Matrix form: Failure of Gaussian Elimination (review):
Example 2:
Example 1:

The overall Pattern of Gaussian Elimination:
A recursive Algorithm that reduces Ax=b to Ux=c

If successful: U will contain n pivots on the diagonal ( unique solution )
If Fails: U will contain at least 1 zero on the diagonal (no or ∞ solutions)

Gaussian Elimination using matrices:
Two Basic operations:
1) Elimination steps to zero out aij
2) Row exchanges to repair tempiojrary failure

BIG IDEA! We can design Matrices (E and P) to do this for us
Elementary Elimination Matrices Eij : Matrix-Matrix Multiplication: C=AB

Column View:

Row View:

Other Examples: Diagonal Matrices
Left Multiplication: DA

Does AD=DA (is matrix multiplication commutative ?)

Gaussian Elimination:
a sequence of Elementary Elimination Matrices

Point: Now we're doing real Linear Algebra!
( Algebra of matrices and vectors, not arithmetic )

Permutation Matrices Again: AB≠BA in general

Gaussian Elimination:
a sequence of Elementary Elimination Matrices
Point: Matrices Do things!

An important Digression:
General rules of Matrix - Matrix Operations
Matrix Shape:

Matrix Matrix Multiplication: C=AB

Examples ( with numbers ):

Properties of Matrix Multiplication:

Theorem: Matrix Mult is associative.
If A,B,C are matrices of appropriate shapes, then A(BC)=(AB)C

Proof (sketch): Show A(Bc)=(AB)c

Another important Digression:
Operation costs of Matrix-vector and Matrix-Matrix multiplication
( order matters !)

An important Digression:
General rules of Matrix-Matrix Operations

Matrix Powers: AP

The matrix Inverse: A-1

Summary :
Gaussian Elimination using Matrices

 Prev Next