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Inverse of a Matrix method - Solving any systems of equation - calculator

You can calculate any system of linear equations, both homogeneous and heterogeneous with any number of unknowns Using the Inverse of a Matrix method.
To get the result, enter the coefficients in empty cells or by clicking on , change to the text field and enter the coefficients separated by a space. You can also enter full equations with unknowns in the text field (click the example below).

You can add or subtract another equation and coefficients using the buttons or position the cursor over a cell in the last row or column and press the arrow in the appropriate direction on the keyboard.

You can move around cells using tabs, spaces or arrows on the keyboard.

In each field of the calculator you can enter decimal numbers, simple fractions, basic mathematical operation, for example: 1.5; 1/2; 5 + 2; 5-2; 5 * 2; 1/2 + 5; 5 * (8 + 2); (5-2) / 2; 1.5e-3; 1.5 (3); (2/5) * (1/2); 5 + 2 * (5 / 3-2); etc.

You can also enter data into the calculator from the result obtained, just grab any matrix from the results and drop it on cells or a text field.

The results are stored in cookies, which means that the result will be saved until you click .
Thanks to the stored results, you can compare the actions using other linear equation calculators available on our website, eg.

Solving System of linear equations by Cramer's method

Solving Systems with Gaussian Elimination calculator

Solving Systems with Gauss-Jordan Elimination calculator

System of equations using the inverse matrix method

Przykład 1:

2x-2y+z=-3
x+3y-2z=1
3x-y-z=2

Przykład 2:

2x-0y+z=2
x-3y-0z=1
x+y-2z=0

Przykład 3:

2x-2y+z-w=7
-x+4y-5z+3w=1
x+y-z+w=4
-4x+2y+z-2w=6

System of equations using the inverse matrix method

decimal fractions , number of decimal places =

Result

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Useful information

A system of linear equations is a system of figures:
$$\large{\left\{\begin{array}{1} \mathrm{a_{11}x_{1} + a_{12}x_{2}}&+&\cdots&+&\mathrm{a_{1n}x_{n} = b_{1}},\\ \mathrm{a_{21}x_{1} + a_{22}x_{2}}&+&\dotsb&+&\mathrm{a_{2n}x_{n} = b_{2}},\\ \vdots&\vdots & \ddots & \vdots \\ \mathrm{a_{m1}x_{1} + a_{m2}x_{2}}&+&\cdots&+&\mathrm{a_{mn}x_{n} = b_{m}} \end{array} \right.}$$
gdzie:

$\mathrm{\large{a_{11},...,a_{mn}}}$ - these are the coefficients of the equation (data)
$\mathrm{\large{b_{1},...,b_{m}}}$- free words (data)
$\mathrm{\large{x_{1},...,x_{n}}}$- unknown

There are two types of systems of linear equations:

• homogeneous systems of linear equations,

• systems of linear equations non-homogeneous,
A system of linear equations is called homogeneous if all the terms are zero.
$$\large{\left\{\begin{array}{1} \mathrm{a_{11}x_{1} + a_{12}x_{2}}&+&\cdots&+&\mathrm{a_{1n}x_{n} = 0},\\ \mathrm{a_{21}x_{1} + a_{22}x_{2}}&+&\dotsb&+&\mathrm{a_{2n}x_{n} = 0},\\ \vdots&\vdots & \ddots & \vdots \\ \mathrm{a_{m1}x_{1} + a_{m2}x_{2}}&+&\cdots&+&\mathrm{a_{mn}x_{n} = 0} \end{array} \right.}$$

The methods of solving linear equations are divided into two classes:

• finite methods - allow to obtain a solution after performing a finite number of arithmetic operations, and the obtained solutions are burdened with rounding errors only.

• iterative methods - they consist in determining the sequence of vectors converging to the solution of the system. The obtained solutions are subject to errors in the method and rounding. These methods, however, allow to determine solutions with any predetermined accuracy.

The method of using the inverse matrix to solve systems of equations is:

1) saving the system of equations in matrix form,

2) computation of the inverse

3) Multiply the calculated inverse with the resulting matrix

4) save the results

The inverse matrix method can only be used in systems of Cramer's linear equations, i.e. those where:

• There are as many equations as there are unknowns

• The main determinant of the system (composed of coefficients on the variables) is not zero
When multiplying on the left (matrix multiplication in general is not commutative) the equation is A·X=B by matrix A^-1, then A^-1·A·X=A^-1·B we get the solution X=A^-1·B.

The inverse matrix of A can be determined in any way:

• using elementary operations,

A^D

To compute the inverse of a non-singular matrix of degree n with only row elementary operations, you reduce the [A|I] block matrix to [I|B] . The resulting matrix B is the matrix we are looking for inverse to A . Symbolically, it can be written:
$$\left[\left.A\right|I\right]\sim\left[\left.I\right|A^{-1}\right] $$

System of linear equations - Wikipedia (EN)
Matrix - Wikipedia (EN)

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