Augmented Matrix

One of the two matrices combined to form an augmented matrix is a coefficient matrix of the variables, and the other contains constants from the equations. If one matrix has m columns and the other has n columns, then the augmented matrix formed has m + n columns. The number of rows in the matrix will be the same as the number of equations.

For example, consider a system of equations:

2x+3y=8
4x-y=2

The coefficient matrix is:
 

The constant matrix is:
    
So, the augmented matrix is:

Augmented matrices provide an efficient way to solve systems of linear equations using row operations. Augmented matrices represent an entire system as one matrix to which row operations can be applied for efficient solutions. They are also foundational for row operations, matrix forms, and understanding the identity matrix and inverse matrices. 

Augmented matrices and coefficient matrices are both used to represent systems of linear equations in matrix form. Both matrices are similar in the following ways:

• Each row represents one equation 
• Each column in the coefficient matrix corresponds to a specific variable. 
• Missing variables in any equation are represented with a zero to keep alignment consistent.

The coefficient matrix only has the coefficients of variables like x and y from each equation. The augmented matrix includes coefficients as well as constant terms.

The properties of augmented matrices help us understand how they are structured, what operations can be used to solve them, and how they show if a system has no solutions, one solution or infinitely many solutions. These properties are:

• An augmented matrix is always rectangular
• Each row in an augmented matrix represents one equation, so the number of rows in the matrix is equal to the number of equations in the system.
• The number of columns in an augmented matrix is equal to the number of variables + 1. The extra column represents constants from the right-hand side of the equation.
• Multiplying a row by any constant value changes the matrix but not the solution of the system represented by it.
• Elementary row operations like swapping or addition/subtraction of one row from another, can be applied to any row of an augmented matrix to solve for systems of equations.
   

Simplifying an augmented matrix helps find the solution to a system of linear equations. This is done using row operations to transform the matrix into reduced row echelon form (RREF) using the Gauss-Jordan elimination method.

Row transformations change the first part of the augmented matrix into the identity matrix. The values of the last column obtained are the solution for the system of linear equations.
Suppose we are given a system of linear equations:

a₁x+b₁y+c₁z=d₁
a₂x+b₂y+c₂z=d₂
a₃x+b₃y+c₃z=d₃

We can represent this system using an augmented matrix as follows:

By applying elementary row operations, we simplify the matrix into row-reduced echelon form (RREF), where the left side becomes the identity matrix:

This final matrix shows that the system has a unique solution, given by:

x=p, y=1, z=r

The values in the last column of the augmented matrix are the values that solve the original system of equations.
 

For finding augmented matrix, the steps are as follows - 

Step 1: At first, the coefficient matrix is formed by the coefficients from each equation. 

Step 2: The constant matrix has to be placed on the right-hand side of each equation. 

Step 3: For Augmented matrix, combine both coefficient matrix and constant matrix side-by-side.
 

An augmented matrix is formed by combining two matrices from a system of linear equations: the coefficient matrix and the constant matrix (column matrix). They are placed side by side and separated by a dotted line.

For instance, for a system of linear equations:

2x+3y-z=5
4x-y+2z=6
-3x+2y+z=-4

The coefficient matrix A is:
   

The constant matrix B is:
  

The augmented matrix is:

For a general system of linear equations with coefficients aij​ and variables x₁, x₂,. . ., x_n

The system can be written as:

a₁₁x₁+a₁₂x₂+ . . . +a_1nx_n=b₁
a₂₁x₁+a₂₂x₂+ . . . +a_2nx_n=b₂
    .                                    .
    .                                    .
    .                                    .
a_m1x₁+a_m2x₂+ . . . +a_mnx_n=b_m

The augmented matrix for this system is:

Augmented matrices are key mathematical tools used in algebraic calculations. Listed below are some real life applications of this tool:

• Solving Electrical Circuit Problems: Augmented matrices are used in electrical engineering to solve systems of equations in Kirchhoff’s law that model complex circuits.
   
• Image transformations and computer graphics: In computer graphics, systems of linear equations describe how images are scaled, rotated, or translated. These systems are frequently solved using augmented matrices, especially when combining transformation matrices with point coordinates.
   
• Help In Navigation: Augmented matrix is used in GPS systems for easy navigation and finding location. It is used in both 2D and 3D planes for calculating distances, positions, and directions. It also simplifies complex geometric and trigonometry issues. 
   
• Economics and business forecasting: Matrices are used for finding economic indicators. It also predicts variables used in supply chain, resource allocation or price equilibrium. 
   
• Balancing chemical reactions: Balancing complex chemical equations typically involves setting up a system of linear equations based on the conservation of atoms. These equations are represented in an augmented matrix, making it easier to solve for stoichiometric coefficients. Stoichiometric coefficients are the numbers placed in front of chemical formulas in a balanced chemical equation.