Matrix Multiplication

Matrix Multiplication is a binary operation that takes a pair of matrices, and produces another matrix. Numbers such as the real or complex numbers can be multiplied according to elementary arithmetic

If A is an n × m matrix and B is an m × p matrix,


the matrix product AB (denoted without multiplication signs or dots) is defined to be the n × p matrix



If the column and rows are in the same size it is defined.



EXAMPLE:

Matrix Subtraction

Matrix Subtraction- you can subtract two matrices (of the same size) by adding their corresponding entries.

EXAMPLES: 
1.

2.
Given the following matrices, find A – Band A – C, or explain why you can not.

A and B are the same size, each being 2 × 3 matrices

A and C are not the same size, since A is 2 × 3 and C is 2 × 2. So this subtraction is not defined.


A – C is not defined, because A and C are not the same size.

Matrix Addition

 Matrix Addition is the operation of adding two matrices by adding the corresponding entries together.












EXAMPLE: 

Equality of Matrices

Two matrices are equal if all three of the following conditions are met:

•  Each matrix has the same number of rows.
•  Each matrix has the same number of columns.
•  Corresponding elements within each matrix are equal.

Consider the three matrices shown below.


If A = B then we know that x = 34 and y = 54, since corresponding elements of equal matrices are also equal.
We know that matrix C is not equal to A or B, because C has more columns.

•  Two equal matrices are exactly the same.
•  If rows are changed into columns and columns into rows, we get a transpose matrix. If the original matrix is A, its transpose is usually denoted by A^' or A^t.
•  If two matrices are of the same order (no condition on elements) they are said to be comparable.
• If the given matrix A is of the order m x n, then its transpose will be of the order n x m. 

1. Column matrix - a matrix that has only one column.
2. Row matrix - a matrix that has only one row.

Analysis of an Electrical Network

EXAMPLE: 

• Applying Kirchhoff's Law to have the Linear equations in the variables.
• Below the linear equations is the Augmented and Gauss-Jordan elimination.

Kirchoff's Law

Kirchhoff's Law is an analysis of such a system uses two properties of electrical networks or two equalities that deal with the current and potential difference in the lumped element model of electrical circuits.

1. All current flowing into a junction must flow out of it.

2. The sum of the products IR ( I is current and  R is resistance ) around a closed path is equal to the total voltage in the path.

EXAMPLE:

a.

b.

c.

d.

Solution: 

Network Analysis

Network Analysis - composed of branches and junctions are used as models in many diverse fields such as economics, traffic analysis, and electrical engineering.

EXAMPLE: