Augmented Matrix Calculator

 The Augmented Matrix Calculator is a powerful tool that assists in solving systems of linear equations using the Gauss-Jordan elimination method. This method simplifies the augmented matrix to find solutions for the variables involved.

Understanding Augmented Matrices

An augmented matrix is formed by combining the coefficients of the variables from a system of linear equations with their corresponding constant terms. The number of rows in the augmented matrix corresponds to the number of equations (or variables) in the system.

Example of an Augmented Matrix

Consider the following system of linear equations:

1. $3x+5y=10$
2. $7x+9y=15$

The augmented matrix for this system can be represented as:

$$\left[\begin{array}{ccc}3& 5& 10\\ 7& 9& 15\end{array}\right]$$

Steps to Solve an Augmented Matrix

To solve an augmented matrix, follow these steps:

Step 1: Normalize the First Row

Divide the first row by the leading coefficient (in this case, 3):

$$R_0=\frac{R_0}{3}⟹\left[\begin{array}{ccc}1& \frac{5}{3}& \frac{10}{3}\\ 7& 9& 15\end{array}\right]$$

Step 2: Eliminate the First Column Below

Multiply the first row by 7 and subtract it from the second row:

$$R_1=R_1-7R_0⟹\left[\begin{array}{ccc}1& \frac{5}{3}& \frac{10}{3}\\ 0& -\frac{8}{3}& -\frac{25}{3}\end{array}\right]$$

Step 3: Normalize the Second Row

Multiply the second row by $-\frac{3}{8}$:

$$R_1=-\frac{3}{8}{R}_1⟹\left[\begin{array}{ccc}1& \frac{5}{3}& \frac{10}{3}\\ 0& 1& \frac{25}{8}\end{array}\right]$$

Step 4: Eliminate Above Using Second Row

Subtract $\frac{5}{3}{R}_1$ from the first row:

$$R_0=R_0-\frac{5}{3}{R}_1⟹\left[\begin{array}{ccc}1& 0& -\frac{15}{8}\\ 0& 1& \frac{25}{8}\end{array}\right]$$

The resulting matrix is now in reduced echelon form, which allows us to read off the solutions for $x$ and $y$.

Properties of Augmented Matrices

An augmented matrix possesses several key properties:

• It combines coefficient and constant matrices.
• The number of rows equals the number of equations.
• It can be transformed into reduced row echelon form to solve for variables.

Input and Output Requirements

To use the augmented matrix calculator effectively, you need to provide:

• Input: Coefficients and constants from your system of equations.
• Output: Solutions for each variable after performing Gauss-Jordan elimination.

With this understanding, you can utilize the augmented matrix calculator to solve various systems of linear equations efficiently.