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Math Calculators ▶ Augmented Matrix Calculator

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**Table of Content**

The free augmented matrix calculator is specially designed to solve an augmented matrix of linear equations.

An augmented matrix is a matrix formed by merging the column of two matrices to form a new matrix.

The augmented matrix is one method to solve the system of linear equations. The number of rows in an augmented matrix is always equal to the number of variables in the linear equation.

Let’s understand the concept of an augmented matrix with the help of three linear equations!

**a1x + b1y + c1z = d1**

**a2x + b2y + c2z = d2**

**a3x + b3y + c3z = d3**

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

Matrix of Constant terms – B = $$\begin{bmatrix}d_1\\d_2\\d_3\end{bmatrix}$$

Matrix of Variables – C = $$ \begin{bmatrix}x\\y\\z\end{bmatrix}$$

Here we are solving the augmented matrix in the example below.

Let us suppose we have the following system of linear equations:

**3 x + 5y = 10**

** 7x + 9 y = 15**

For instant calculations, you may better use the augmented matrix calculator 2×3. but we will also consider the manual calculations here:

In the example below, All the steps are explained in detail.

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

Divide row zeroth row by 3: R0 = R0/3

$$ \left[ \begin{array}{cc|c}1& \frac{5}{3}&\frac{10}{3}\\ 7&9&15 \\ \end{array}\right]$$

Multiply the zeroth by 7 and subtract the row first by the zeroth and multiply the zeroth by 7 R0: R1 = R1 – 7R0

$$ \left[ \begin{array}{cc|c}1&\frac{5}{3}&\frac{10}{3}\\0& \frac{-8}{3}& \frac{-25}{3}\\ \end{array} \right]$$

Multiply first row 1 by 3/-8: R1 = 3/-8 R1

$$ \left[ \begin{array}{cc|c}1& \frac{5}{3} & \frac{10}{3} \\0&1&\frac{25}{8} \\ \end{array}\right] $$

Multiplied first row by 5/3 and subtract it from zeroth rowR1: R0 = R0 – 5/3R1

$$ left[\begin{array}{cc|c}1&0&\frac{-15}{8} \\0&1&\frac{25}{8}\\\end{array}\right]$$

The reduced echelon form of the matrix is also considered augmented matrix.

Augmented matrix possesses the following properties:

- The variables in the linear equations and the constant term determine the number of columns.
- The number of systems of equations is the same as the number of rows.
- The augmented matrix’s rows can be swapped around.
- A constant can be used to multiply or divide the elements of a certain row.
- The specific row of the matrix can be added to and removed from other rows.
- A matrix row’s multiple can be applied to another matrix row.

Our augmented matrix solver requires the following inputs to generate an accurate result.

**Input:**

- Set the order of the matrix
- Enter the elements of the matrix
- Hit the calculate button

**Output:**

- Detailed steps of the augmented matrix represented
- Solution of the linear equation

A simple matrix only represents the coefficients of the linear equation, but the augmented matrix represents the coefficients as well as the constant resultant values of the linear equations.

From the source Wikipedia: Augmented matrix, The inverse of a matrix

From the source of tutorial.math.lamar.edu: Elementary row operations, Augmented Matrices