\[ \left( \begin{array}{cccc} 1 & 2 & 1 & 2 \\ 2 & 1 & 2 & 1 \\ 3 & 2 & 3 & 2 \\ 3 & 3 & 3 & 3 \\ 5 & 3 & 5 & 3 \end{array} \right) \]
To reduce this matrix to row reduced echelon form perform elementary row operations - adding or subtracting multiples of each row, interchanging rows, or scaling rows - until we have a matrix satisfying
1. All zero rows are at the bottom of the matrix.
2. The leading entry of each nonzero row after the first occurs to the right of the leading entry of the previous row.
3. The leading entry in any nonzero row is 1.
4. All entries in the column above and below a leading 1 are zero.
Subtract two times row 1 from row 2, subtract 3 times row 1 from rows 3 and 4, and subtract 5 times row 1 from row 5. We get
\[ \left( \begin{array}{cccc} 1 & 2 & 1 & 2 \\ 0 & -3 & 0 & -3 \\ 0 & -4 & 0 & -4 \\ 0 & -3 & 0 & -3 \\ 0 & -7 & 0 & -7 \end{array} \right) \]
Divide row 2 by -3
\[ \left( \begin{array}{cccc} 1 & 2 & 1 & 2 \\ 0 & 1 & 0 & 1 \\ 0 & -4 & 0 & -4 \\ 0 & -3 & 0 & -3 \\ 0 & -7 & 0 & -7 \end{array} \right) \]
Subtract times row 2 from row 1, add four times row 2 to row 3, add three times row 2 to row 4 and add seven times row 2 to row 5.
\[ \left( \begin{array}{cccc} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right) \]
This is the row reduced echelon form of the matrix. Row reduced echelon form is also called Hermitian normal form.