System of equations into matrix form calculator

Here you can solve systems of simultaneous linear equations using Gauss-Jordan Elimination Calculator with complex numbers online for free with a very detailed solution. Our calculator is capable of solving systems with a single unique solution as well as undetermined systems which have infinitely many solutions. In that case you will get the dependence of one variables on the others that are called free. You can also check your linear system of equations on consistency using our Gauss-Jordan Elimination Calculator.

Have questions? Read the instructions.

About the method

To solve a system of linear equations using Gauss-Jordan elimination you need to do the following steps.

1. Set an augmented matrix.
2. In fact Gauss-Jordan elimination algorithm is divided into forward elimination and back substitution. Forward elimination of Gauss-Jordan calculator reduces matrix to row echelon form. Back substitution of Gauss-Jordan calculator reduces matrix to reduced row echelon form. But practically it is more convenient to eliminate all elements below and above at once when using Gauss-Jordan elimination calculator. Our calculator uses this method.
3. It is important to notice that while calculating using Gauss-Jordan calculator if a matrix has at least one zero row with NONzero right hand side (column of constant terms) the system of equations is inconsistent then. The solution set of such system of linear equations doesn't exist.

To understand Gauss-Jordan elimination algorithm better input any example, choose "very detailed solution" option and examine the solution.

Instructions: Use this online calculator to get a system of linear equations from its matrix representation, showing all the steps. First, click on one of the buttons below to specify the dimension of the matrix representation, then you need to specify \(A\) and \(b\).

For each of the matrix and vector, click on the first cell and type the value, and move around the matrix by pressing "TAB" or by clicking on the corresponding cells, to define ALL the matrix values.

More about this matrix for to system of equations calculator.

Often times you will have a system in matrix form, with \(Ax = b\) and you will want to actually express the matrix form into the regular linear equation form, just to see the equations in a more clear way.

If you are provided with matrix form, perhaps you will like to solve the system using Cramer's Rule, or you maybe want to solve it by using the inverse method.

Why would you pass from matrix form to system of equation forms

The two forms are completely interchangeable, but perhaps the system of equation form allows you for a more clear interpretation of the situation you are facing, particularly in cases where the setting of the linear equation is tied to real variables.

How to convert a matrix form into system of equations form

Simple. You need to take a look at the matrix \(A\), row by row. Each row of \(A\) corresponds to an equation. Now, each column of those rows is associated to a certain variable.

What happens when a coefficient is zero? In that case, the associated variable does not appear in the equation.