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Knowing how to calculate the inverse of a matrix is very useful to easily solve systems of linear equations using the matrix inversion method. For this reason, we put in your hands this Inverse Matrix Calculator with which you can practice, study and fully understand how to obtain the inverse of a matrix, thanks to the fact that the solutions are explained step by step using different methods.

To calculate the inverse of a matrix, you only have to do three simple steps:

- The first step is to introduce the matrix to which you are going to calculate its inverse. You can modify the number of rows or columns according to the dimension of the matrix that you need to enter using the
**+–**buttons. - In the second step you will have to choose if you want to obtain the solution expressed in decimal numbers or not.
- And finally, you just have to press the button «Calculate». When you have pressed said button, a box will automatically be displayed with the solution explained step by step through the use of different methods.

The inverse of a matrix is another matrix of equal dimensions and that if multiplied by the original matrix results in the identity matrix. In other words, the inverse of the matrix [A], designated as [A]^{–1}, is defined by the following property:

[A]·[A]^{–1}=[A]^{–1}·[A]=[I]

where [I] is the identity matrix.

You should keep in mind that only square matrices can have an inverse matrix, in other words, a square matrix can be an **invertible matrix**. This is because the definition of an inverse matrix is based on the concept of identity matrix [I], and only square matrices have an identity matrix associated with it.

A matrix whose determinant is equal to 0 is a non-invertible matrix, which is why this type of matrix is called a singular matrix.

- A non-singular square matrix has only a single inverse matrix.
- The square matrix A is an invertible matrix, only if its determinant is a nonzero value, |A| ≠ 0.
- If A and B are non-singular matrices, then the product AB results in a non-singular matrix and (A·B)
^{-1}= B^{-1}·A^{-1} - If A is not singular, then the inverse matrix of the transpose of A is equal to the transpose of the inverse of A, (A
^{T})^{-1}= (A^{-1})^{T}. - If the product of matrices A and B is equal to the identity matrix, then the matrices are inverses of each other.

There are several methods to find the inverse matrix, but the general method to calculate the inverse of a matrix consists of using the following formula:

where |A| is the determinant of A and Adj(A) is the adjoint matrix of A. We will illustrate how to use the previous formula by calculating the inverse of a 2×2 matrix.

Find 2x2 matrix inverse according to the formula:

${A}^{-1}={\underset{\text{}}{\overline{)\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)}}}^{\left(-1\right)}=\frac{1.00}{\left|A\right|}\times \underset{\text{}}{\overline{)\left(\begin{array}{cc}{C}_{11}& {C}_{21}\\ {C}_{12}& {C}_{22}\end{array}\right)}}=\frac{1.00}{a\times d-b\times c}\times \underset{\text{}}{\overline{)\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)}}$

${\underset{\text{}}{\overline{)\left(\begin{array}{cc}5.00& 8.00\\ 2.00& -3.00\end{array}\right)}}}^{\left(-1\right)}=\frac{1.00}{5.00\times \left(-3.00\right)-8.00\times 2.00}\times \underset{\text{}}{\overline{)\left(\begin{array}{cc}-3.00& -8.00\\ -2.00& 5.00\end{array}\right)}}=\underset{\text{}}{\overline{)\left(\begin{array}{cc}0.10& 0.26\\ 0.06& -0.16\end{array}\right)}}$ ${A}^{-1}={\underset{\text{}}{\overline{)\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)}}}^{\left(-1\right)}=\frac{1.00}{\left|A\right|}\times \underset{\text{}}{\overline{)\left(\begin{array}{cc}{C}_{11}& {C}_{21}\\ {C}_{12}& {C}_{22}\end{array}\right)}}=\frac{1.00}{a\times d-b\times c}\times \underset{\text{}}{\overline{)\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)}}$

The inverse of a matrix is a mathematical concept with numerous applications in a variety of fields. In this article, we’ll explore some of the ways that the calculation of matrix inverse is used in science, engineering, and other areas.

One of the most common applications of matrix inverse is in the field of linear algebra. In this context, the inverse of a matrix is used to solve systems of linear equations. Given a system of equations in the form Ax = b, where A is a matrix, x is a vector of unknowns, and b is a vector of constants, the inverse of A can be used to find the values of x that satisfy the system. This is done by multiplying both sides of the equation by A^{-1}, the inverse of A:

A^{-1} Ax = A^{-1} b

Since A^{-1} A = I, the identity matrix, we can simplify the equation to:

x = A^{-1} b

Thus, the inverse of A allows us to find the solution to the system of equations.

Matrix inverse is also important in the field of engineering, particularly in the analysis of structural systems. In this context, the inverse of a matrix is used to find the internal forces and moments acting on a structure, given the applied loads. This is done by solving a system of equations that relates the loads and the internal forces. The inverse of the stiffness matrix, which relates the internal forces to the deformations of the structure, is used to find the internal forces from the deformations.

In addition to its applications in linear algebra and engineering, matrix inverse is also used in other areas such as statistics, where it is used to find the variance-covariance matrix of a set of random variables, and computer graphics, where it is used to transform objects in a 3D scene.

Overall, the calculation of matrix inverse is a crucial tool in a variety of fields, and its applications are numerous and varied. Understanding how to calculate and use matrix inverse can be essential for solving problems and making decisions in many different fields.