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Augmented Matrix Calculator

Select and enter the values in the augmented matrix calculator, and it will find the solution.

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The augmented matrix calculator solves systems of linear equations using the Gauss-Jordan elimination method.

What Is an Augmented Matrix?

An augmented matrix is formed by combining the coefficient matrix and the constants column of a system of linear equations. It provides a compact way to solve the system using matrix operations. The number of rows in the augmented matrix equals the number of equations (or variables) in the system.

Consider a system of three linear equations:

a₁x + b₁y + c₁z = d₁

a₂x + b₂y + c₂z = d₂

a₃x + b₃y + c₃z = d₃

Matrix Form

Coefficient Matrix (A):

$$ A = \begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix} $$

Constants Column (B):

$$ B = \begin{bmatrix} d_1 \\ d_2 \\ d_3 \end{bmatrix} $$

Variables (X):

$$ X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} $$

The augmented matrix is denoted as [A | B].

How to Solve an Augmented Matrix?

We solve the system using Gauss-Jordan elimination, which reduces the augmented matrix to Reduced Row Echelon Form (RREF).

Example:

Solve the system:

3x + 5y = 10

7x + 9y = 15

Step 0: Write the augmented matrix:

$$ \begin{bmatrix} 3 & 5 & 10 \\ 7 & 9 & 15 \end{bmatrix} $$

Step 1: Make the first pivot 1 by dividing the first row by 3:

$$ R_1 = \frac{R_1}{3} \Rightarrow \begin{bmatrix} 1 & 5/3 & 10/3 \\ 7 & 9 & 15 \end{bmatrix} $$

Step 2: Eliminate the x-term in row 2:

$$ R_2 = R_2 - 7R_1 \Rightarrow \begin{bmatrix} 1 & 5/3 & 10/3 \\ 0 & -8/3 & -25/3 \end{bmatrix} $$

Step 3: Make the pivot in row 2 equal to 1:

$$ R_2 = \left(-\frac{3}{8}\right) R_2 \Rightarrow \begin{bmatrix} 1 & 5/3 & 10/3 \\ 0 & 1 & 25/8 \end{bmatrix} $$

Step 4: Eliminate the y-term in row 1:

$$ R_1 = R_1 - \frac{5}{3}R_2 \Rightarrow \begin{bmatrix} 1 & 0 & -15/8 \\ 0 & 1 & 25/8 \end{bmatrix} $$

The resulting RREF matrix gives the solution:

x = -15/8, y = 25/8

Properties of Augmented Matrices

Number of columns = number of variables + 1 (for constants).
Number of rows = number of equations.
Rows can be swapped without changing the solution.
A row can be multiplied or divided by a non-zero constant.
A multiple of one row can be added to another row.

How Gauss-Jordan Elimination Calculator Works

Input:

Select the order of the matrix (number of equations × number of variables).
Enter all elements of the augmented matrix.
Click "Calculate".

Output:

Step-by-step row operations for elimination.
Final solution of the system of linear equations.

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