System of linear equations calculator

  About system of linear equations calculator

This is a free online system of linear equations calculator with complete, detailed, step-by-step description of solutions, that performs operations with matrices up to 99x99 in size with matrix elements of this type: decimal numbers, fractions, complex numbers, variables.

To start the calculation, you need to first enter the size of the matrix in the input field that you can find from the very top of the screen, also there you can choose the desired method of calculation.

A little below you will find a matrix window in which you need to enter matrix elements using the keyboard. The matrix control panel is also located here, which simplifies work with matrices and contains the following control elements:

• The first element allows you to expand the matrix window. This can be especially useful in cases where you need to perform calculations with very large matrices that do not fit completely. If the matrix is still not visible after expanding the window, you can change the scale of the matrix using the + / - buttons;
• The second element performs the function of copying the matrix input to the memory buffer. This can be useful in cases where you often use the same matrix for calculations, or if you need to move matrices between operations;
• And the last element inserts the previously copied matrix, which allows you to speed up the process of entering the matrix to just a few clicks, instead of doing it manually;

And further down you will find a toolbar that allows you to customize the calculator and make it easier to work with it. It is visually divided into three parts, each of which is responsible for the following functionality:

• The first allows you to select the number format when the solution result is displayed. Also, here you can turn off comments to the solution of the problem if you have already understood how to solve this problem, and you use the calculator to speed up or check your own calculations. Or you can turn off the step-by-step solution entirely if you only need the result of the solution;
• The second contains buttons that allow you to change the type of the matrix input field, erase its elements or the entire matrix, and the largest button with an equal sign, which will take you to the screen with the solution of the problem. All these buttons are duplicated by keys on the keyboard. To find out which key on the keyboard to press, simply hover over one of the buttons and a tooltip will appear with the name of the key. You can also use the arrow keys on your keyboard to move the cursor between matrix input fields;
• And the last one allows you to choose the number of digits after the decimal point for rounding non-integer numbers. Also, here you can immediately see an example of how rounded fractions will look;

  What is system of linear equations?

A system of linear equations is a set of two or more linear equations with the same variables. Solving a system of linear equations means finding these variables.

  How to solve a system of linear equations using Gaussian elimination?

We need to write a system of linear equations in matrix form and then using Gaussian elimination we can bring this matrix to the row echelon form. After that, in the last row in the column of free coefficients, we get the last root of the system, then using the Back Substitution, we find all the other roots of the system.

  How to solve a system of linear equations using Cramer's rule?

Kramer's rule for solving systems of linear equations involves first finding the determinant of the matrix of coefficients of the system of linear equations. Next, we need to form a new matrix based on the matrix of coefficients, but instead of the first column, put a column of the free coefficient there, then we need to find the determinant of this matrix and divide it by the determinant of the matrix of coefficients, and the result will give us the first root. Next, similarly to the first root, we need to find the rest of the roots by substituting the column with free coefficients in the matrix with coefficients instead of the second, third column, and so on until the last column.

  How to solve a system of linear equations using Gauss-Jordan method?

We need to apply the Gauss-Jordan method to the matrix form of the system of linear equations and then the left side of the matrix becomes the identity, and on the right side we get the roots of the system of linear equations.

  How to solve a system of linear equations using Inverse matrix method?

First, we need to find the inverse matrix of the matrix of coefficients of the system of linear equations, and then multiply it by the column of free coefficients.

  How to solve a system of linear equations using Bareiss algorithm?

We need to apply the Bareiss algorithm to the matrix form of the system of linear equations and then the left side of the matrix becomes the identity, and on the right side we get the roots of the system of linear equations.