# Inverse of a Matrix theory

Inverse of a Matrix

Definition method.

If the product between two matrices is the identity matrix, then we say that the matrices are “inverse”; because by multiplying them we obtain the neutral element for the product .

Consider the inverse matrix: Bioprofe | Calculate the inverse of a matrix|01

It must accomplish that: A · A-1 = I

Observe the following example. Bioprofe | Calculate the inverse of a matrix|02 Bioprofe | Calculate the inverse of a matrix|03 Bioprofe | Calculate the inverse of a matrix|04

So, we obtain the following equations system to get the inverse matrix values. Bioprofe | Calculate the inverse of a matrix|05 Bioprofe | Calculate the inverse of a matrix|06

Determinant method.

By applying this method we say that: Bioprofe | Calculate the inverse of a matrix|07

Where att (A)t is attached matrix and det (A) is the determinant of the matrix A.

The A determinant solution can be seen in the corresponding section about determinants theory.

att (A)t is obtained as follows:

Given matrix A: Bioprofe | Calculate the inverse of a matrix|08

Its attached is formed by the determinings which are result by suppressing a row and column of each of one of the elements.

For example, the a32 element position would be occupied by the determining which results by eliminating the third row and second column, that is: Bioprofe | Calculate the inverse of a matrix|09

and as the row + column (3+2) is odd we change the sign. Bioprofe | Calculate the inverse of a matrix|10

Then we perform its transpose: Bioprofe | Calculate the inverse of a matrix|11