# Examples of Gauss-Jordan elimination

$3 \times 3$ example

Find the inverse matrix of a 3x3 matrix,

, using Gauss-Jordan elimination.

**Sample answer**
To obtain the inverse matrix,
we define
a matrix in which the matrix $A$ and the identity matrix $I$ are arranged side by side,

. This matrix is called augmented matrix.
We transform the matrix $A$ in the augumented matrix to the identity matrix $I$ by performing elementary row operations, i.e.,

.
As a result, the identity 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

$4 \times 4$ example

Find the inverse matrix of a 4x4 matrix,

, using Gauss-Jordan elimination.
.

**Sample answer**
To obtain the inverse matrix,
we define
a matrix in which the matrix $A$ and the identity matrix $I$ are arranged side by side,

. This matrix is called augmented matrix.
We transform the matrix $A$ in the augumented matrix to the identity matrix $I$ by performing elementary row operations, i.e.,

.
As a result, the identity 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