Operations with Matrices

You can represent matrices in any of the following:
1) A matrix can be denoted by an uppercase letter such as,
A, B, C, . . .

2) A matrix can be denoted by a representative element enclosed in brackets, such as,

3) A matrix can be denoted by a rectangular array of numbers.

Equality of Matrices: 
If two matrices A= (Aij) and B= (Bij) have the same size mxn and Aij = Bij ; then A=B. 

*It is important to know the size of your matrices so that you'll know whether you can perform the operation or not. 

*Matrices A and B are not equal because they have different sizes. Also with Matrices C and D, only if 
x = 3. 








* Column Matrix or Column Vector
A matrix with only 1 column. 

*Row Matrix or Row Vector
A matrix with only 1 row. 

MATRIX ADDITION
*You can only add matrices of the same size by adding their corresponding entries.  

Definition of a Matrix Addition: 
If A = (aij) and B = (bij) are matrices of size m x n, then their sum is the m x n matrix given by A + B = (aij+ Bij). 

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). 

Network Analysis

The process of finding the voltages across, and the currents through, every component in the network. Networks are composed of networks and junctions that are used in many models in different fields of study. In says that, the total flow into a junction is equal to the total flow out of the junction.

Example:

Polynomial Curve Fitting

*When you are asked to find the polynomial function of degree n-1 and site where it passes through the graph. If all the x-coordinates of the points are distinct, then there is precisely one polynomial function of degree n-1.

Example: 

Homogeneous Systems of Linear Equations

*It must have at least one solution. It is also called Trivial/Obvious Solution. If all the variables in a homogeneous system have the value zero, then we can say that it is a Homogeneous Solution. 

Example #1. 

Example #2. 

Gauss-Jordan Elimination and Reduced-Row Echelon Form

Also a method of solving linear equations. Done by transforming the system's augmented matrix into Reduced-Row Echelon form by the use of Elementary Row Operations. 

*Named for Carl Friedrich Gauss and Wilhelm Jordan. Started as Gaussian Elimination and improved by Wilhelm Jordan. 

Reduced-Row Echelon Form: 
The following are the conditions that your matrix is in RREF: 
1. When it satisfies all the conditions in Row-Echelon Form. 

Gaussian Elimination Method with Back Substitution

The method of solution where in it is based on addition/elimination, there is a systematized method for solving the three-or-more variable system. This method is called "Gaussian Elimination" (with the equations ending up in what is called "row-echelon form").


Elementary Row Operations:

1. Interchange two equations.

2. Multiply an equation by a nonzero constant.

3. Add a multiple of an equation to another equation.

Row-Echelon Form of a Matrix

*The term "echelon" refers to the stair-step pattern formed by the nonzero elements of the matrix. 

Definition of Row-Echelon Form of a Matrix: 

A Matrix that is in Row-Echelon Form has the following properties: 

1. All rows consisting entirely of zeros occur at the bottom of the matrix. 

2. For each row that does not consist entirely of zeros, the first nonzero entry is 1 (called a leading 1). 

3. For two successive (nonzero) rows, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row. 

Gaussian Elimination w/ Back Substitution: 

1. Write the augmented matrix of the system of linear equations. 

2. Use the elementary row operations to rewrite the augmented matrix in row-echelon form. 

3. Write the system of linear equations corresponding to the matrix in row-echelon form, and use back substitution to find the solution. 

Example #1. 

Example #2. 

Matrix

*This is a rectangular array of real or complex numbers, symbols, expressions arranged in rows and columns. 

Entry/Entries
*aij
*The individual items in a matrix, wherein (i) is the row and (j) is the column. 

Rows
*Horizontal lines. 

Columns
*Vertical lines.

 Real Matrix
*A entry consisting of real numbers. 

Square Matrix
*A matrix with the same number of rows and columns. 

Sizes of Matrices:
*A matrix with m rows and n columns (an m x n matrix) is said to be of size m x n.  

*One common use of matrix is to represent systems of linear equations. 

Augmented Matrix
*The matrix derived from the coefficients and constant terms of a system of linear equations. 

Coefficient Matrix
*The matrix containing only the coefficients of the system.