Row operations provide a systematic way to calculate the inverse of a square matrix when one exists.

Augment with the identity

Place the matrix beside an identity matrix of the same size:

$$[A\mid I].$$

Then perform elementary row operations on the entire augmented matrix. If the left side can be reduced to the identity,

$$[A\mid I]\longrightarrow[I\mid A^{-1}],$$

the right side has become the inverse.

Example

For

$$A=\begin{pmatrix}1&1\0&2\end{pmatrix},$$

start with

$$\left[\begin{array}{cc|cc} 1&1&1&0\ 0&2&0&1 \end{array}\right].$$

Dividing the second row by $2$ and then subtracting the new second row from the first gives

$$\left[\begin{array}{cc|cc} 1&0&1&-1/2\ 0&1&0&1/2 \end{array}\right].$$

Therefore

$$A^{-1}=\begin{pmatrix}1&-1/2\0&1/2\end{pmatrix}.$$

Detecting a noninvertible matrix

If row reduction cannot turn the left side into the identity because a pivot is missing, then $A$ is not invertible. In that case the process exposes the obstruction rather than producing an inverse.

Checking the result

An inverse can be verified by multiplication:

$$AA^{-1}=I$$

and, for a square inverse, equivalently $A^{-1}A=I$. Row reduction is the procedure; multiplication confirms that the resulting matrix really undoes $A$.