Matrices

Definition of Equality of Matrices
Two Matrices A = [aij] and B = [bij] are equal if they have the same size (m x n) and aij = bij
for i and j are less than or equal to m but greater or equal to one.

Matrix Addition
If A = [aij] and B = [bij] are matrices of m x n, then their sum is the m x n matrix given by
                     A + B = [aij + bij]
The sum of two matrices of different sizes is undefined.
Examples

Scalar Multiplication
If A = [aij] is an m x n matrix and c is a scalar, then the scalar multiple of A by c is the m x n matrix given by
          cA = [caij]
Example

Matrix Multiplication

Network Analysis

Polynomial Curve Fitting

Homogeneous System of Linear Equations

A system is said to be homogeneous if all of the constant terms are zero. For a example, a homogeneous system of m equations in n variables has the form:


Example

Gauss-Jordan Elimination

Gaussian Elimination with Back Substitution

We will use Gaussian Elimination to solve the linear system

x1 2x1 3x1 + − 2x2 x2 + + − 3x3 x3 x3 = = = 9 8 3  .

The augmented matrix is

   1 2 3 2 −1 0 3 1 −1         9 8 3     

The Gaussian Elimination algorithm proceeds as follows:

1 2 3 2 −1 0 3 1 −1         9 8 3		−	1 0 0 2 −5 −6 3 −5 −10         9 −10 −24	(Row 1) (Row 2−2Row 1) (Row 3−3Row 1)
		−	1 0 0 2 1 −6 3 1 −10         9 2 −24	(Row 1) (−15Row 2) (Row 3)
		−	1 0 0 2 1 0 3 1 −4         9 2 −12	(Row 1) (Row 2) (Row 3+6Row 2)
		−	1 0 0 2 1 0 3 1 1         9 2 3	(Row 1) (Row 2) (−14Row 3)

We have brought the matrix to row-echelon form. The corresponding system

x1 + 2x2 x2 + + 3x3 x3 x3 = = = 9 2 3  

is easily solved from the bottom up:

x3=3 x2+3=2−x2=−1 x1+2(−1)+3(3)=9−x1=2  

Thus, the solution of the original system is x1=2x2=−1x3=3