To find the rank of a matrix using row operations, we can use the concept of echelon form or reduced row echelon form.

  Here is the step-by-step procedure:

  1. Start with the given matrix.

  2. Perform row operations (such as swapping rows, adding/subtracting rows, or multiplying rows by a non-zero scalar) to bring the matrix into row-echelon form.

   - Row-echelon form has the following properties:

   - The first non-zero entry in each row, called a pivot element, is to the right of the pivot element of the row above.

   - All entries below each pivot element are zero.

  3. Once the matrix is in row-echelon form, count the number of non-zero rows. This will give us the rank of the matrix.

   - The rank of a matrix is the number of linearly independent rows or columns in the matrix.

  Note that row operations do not change the row space of the matrix. So, row-echelon form and the original matrix have the same row space, and thus, the same rank.

  It's important to mention that if we bring the matrix into reduced row echelon form instead of row-echelon form, the number of non-zero rows will still give us the rank of the matrix. Reduced row echelon form has the additional property that each pivot element is equal to 1, and the entries above and below each pivot are zero.

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