Inverse matrix to solve system of equations calculator

The one famous method of solving systems of linear algebraic equations (SLAE) is the inverse matrix method. Suppose we have SLAE of two equations with two unknowns.

a11xa12 yb1a21xa22yb2

Intoduce the notations: A - SLAE's matrix of the form:

Aa11a12a21a22

X - column vector of unknowns which are to be found:

Xxy

B - vector column of free coefficients:

Bb1b2

So, the initial SLAE can be rewritten in matrix notations:

AXB

In order to solve this matrix equation, multiply both its sides from the left by A-1 matrix:

A1AXA1B

Here, A-1 - is inverse matrix of matrix A. Such matrix exists for any square nondegenerate matrix (i.e. its determinant doesn't equal to zero).

The conditions above show the boundaries of application of inverse matrix method for SLAE solution. First of all: SLAE's matrix A must be square. This means, that the number of equations must be equal to the number of variables. In the second place: the determinant of matrix A must not be equal to zero:

A0

In addition, the inverse matrix shares wonderful feature: its product with initial matrix is commutative and equals to the identity matrix:

A1AAA1E

Returning to the solution of our matrix equation, we get:

EXXA1B

So, in order to solve SLAE by inverse matrix method, first of all one need to check, that inverse matrix does exist and then find it and multiply by the vector B.

The purpose of our online calculator is to solve SLAE by inverse matrix method. The calculator finds the step by step solution. The SLAE's equations are entered in their natural form. The equation's coefficient might be not only numbers and fractions, but also the parameters. In the latter case the calculator gives solution in the common form.

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Solve the system of linear equations step by step

This calculator will solve the system of linear equations of any kind, with steps shown, using either the Gauss-Jordan elimination method, the inverse matrix method, or Cramer's rule.

Related calculator: System of Equations Calculator

Comma-separated, for example, x+2y=5,3x+5y=14.

Leave empty for autodetection or specify variables like x,y (comma-separated).

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Your Input

Solve $$$\begin{cases} 5 x - 2 y = 1 \\ x + 3 y = 7 \end{cases}$$$ for $$$x$$$, $$$y$$$ using the Gauss-Jordan Elimination method.

Solution

Write down the augmented matrix: $$$\left[\begin{array}{cc|c}5 & -2 & 1\\1 & 3 & 7\end{array}\right]$$$.

Perform the Gauss-Jordan elimination (for steps, see Gauss-Jordan elimination calculator): $$$\left[\begin{array}{cc|c}5 & -2 & 1\\0 & \frac{17}{5} & \frac{34}{5}\end{array}\right]$$$.

Back-substitute:

$$$y = \frac{\frac{34}{5}}{\frac{17}{5}} = 2$$$

$$$x = \frac{1 - \left(-2\right) \left(2\right)}{5} = 1$$$

Answer

$$$x = 1$$$A

$$$y = 2$$$A

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