the inverse of the input matrix.

For example, take the following matrix:

The first step to compute its inverse is to create the augmented matrix

Call the first row of this matrix and the second row . Then, add row 1 to row 2 This yields

Next, subtract row 2, multiplied by 3, from row 1 which yields

Finally, multiply row 1 by -1 and row 2 by 2 This yields the identity matrix on the left side and the inverse matrix on the right:

Thus, It works because the process of Gaussian elimination