Finding Inverse Matrix by Elementary Operations (Transformation)

An elementary operation (or transformation) is one of three simple moves you can make on the rows (or, separately, the columns) of a matrix: interchanging two rows, denoted $R_i \leftrightarrow R_j$; multiplying every element of a row by a non-zero scalar; or adding a scalar multiple of one row to another row, $R_i \to R_i + kR_j$. The same three moves apply to columns, written with $C_i$ in place of $R_i$.

To find the inverse of an invertible matrix $A$ using this method, write the equation $A = IA$ and apply a sequence of row operations to both sides simultaneously, continuing until the left-hand side becomes the identity matrix — at that point, the right-hand side has become $A^{-1}$:

(The mirror version, starting from $A = AI$ and using column operations throughout, works the same way.) One rule must be respected: once you commit to row operations, use only row operations for the rest of that computation — mixing rows and columns midway invalidates the result. …