A simple example of finding the inverse matrix of a 3x3 matrix, using Gauss-Jordan elimination
Find the inverse matrix of a 3x3 matrix,
, using Gauss-Jordan elimination.
To obtain the inverse matrix,
a matrix in which the matrix $A$ and the unit matrix $I$ are arranged side by side,
. This matrix is called augmented matrix.
We transform the matrix $A$ in the augumented matrix to the unit matrix $I$ by performing elementary row operations, i.e.,
As a result, the unit matrix in the right half of the augmented matrix becomes the inverse of $A$.
This method of finding the inverse matrix is called Gauss-Jordan elimination.
(Here, the dotted line drawn vertically is merely a convenience for distinguishing between the left side and the right side.)
According to this method, we perform elementary row operations as follows.
Therefore, we obtain