# Matrix Inverse

The multiplication of a matrix by its inverse is equal to the identity matrix.

**A · A ^{-1} = A^{-1} · A = I**

#### Properties of the Inverse Matrix

**(A · B) ^{-1} = B^{-1} · A^{-1} **

**(A ^{-1})^{-1} = A **

**(k · A) ^{-1} = k^{-1} · A^{-1} **

**(A ^{t})^{-1} = (A^{-1})^{t} **

#### Steps to Calculate the Inverse Matrix

A is a square matrix of order n. To calculate the inverse of A, denoted as **A ^{-1}**, follow these steps:

1 Construct a matrix of type** M = (A | I)**, that is to say, *A* is in the left half of *M* and the identity matrix * I* is on the right.

Consider an arbitrary 3x3 matrix:

Place the identity matrix of order 3 to the right of *Martix M*.

2Using the Gaussian elimination method, transform the left half, **A**, to the identity matrix, located to the right, and the matix that results in the right side will be the inverse of matrix: A^{-1}.

**r _{2} - r_{1} **

**r _{3} + r_{2}**

**r _{2} - r_{3} **

**r _{1} + r_{2}**

**(−1) · ****r _{2} **

The inverse matrix is:

#### Examples

Calculate the matrix inverse of:

1 Construct a matrix of type M = (A | I).

2Using the Gaussian elimination method, transform the left half, A, in the identity matrix, located to the right, and the matix that results in the right side will be the inverse of matrix: A^{-1}.

For what values of m in the matrix does not support an inverse?

For any real value of m, there is the inverse A^{-1}.