Editing Rank (linear algebra) (section)

=== Rank from row echelon forms ===
{{main|Gaussian elimination}}
A common approach to finding the rank of a matrix is to reduce it to a simpler form, generally [[row echelon form]], by [[elementary row operations]]. Row operations do not change the row space (hence do not change the row rank), and, being invertible, map the column space to an isomorphic space (hence do not change the column rank). Once in row echelon form, the rank is clearly the same for both row rank and column rank, and equals the number of [[Pivot_element|pivots]] (or basic columns) and also the number of non-zero rows.

For example, the matrix {{mvar|A}} given by
<math display="block">A=\begin{bmatrix}1&2&1\\-2&-3&1\\3&5&0\end{bmatrix}</math>
can be put in reduced row-echelon form by using the following elementary row operations:
<math display="block">\begin{align}
\begin{bmatrix}1&2&1\\-2&-3&1\\3&5&0\end{bmatrix}
&\xrightarrow{2R_1 + R_2 \to R_2}
\begin{bmatrix}1&2&1\\0&1&3\\3&5&0\end{bmatrix}
\xrightarrow{-3R_1 + R_3 \to R_3}
\begin{bmatrix}1&2&1\\0&1&3\\0&-1&-3\end{bmatrix} \\
&\xrightarrow{R_2 + R_3 \to R_3}
\,\,
\begin{bmatrix}1&2&1\\0&1&3\\0&0&0\end{bmatrix}
\xrightarrow{-2R_2 + R_1 \to R_1}
\begin{bmatrix}1&0&-5\\0&1&3\\0&0&0\end{bmatrix}~.
\end{align}</math>
The final matrix (in reduced row echelon form) has two non-zero rows and thus the rank of matrix {{mvar|A}} is 2.