In linear algebra, when calculating the rank of a matrix, we often reduce it to row echelon form or reduced row echelon form (RREF). The main goal is to perform row operations to make the lower triangle (below the diagonal) zeros, converting the matrix into an upper triangular form. Here's how it works:

Steps to Convert a Matrix to Row Echelon Form (Upper Triangle of Zeros):

1. Start with the leftmost pivot column: Identify the first non-zero column (pivot column) and ensure the first entry (pivot) in this column is non-zero. If necessary, swap rows.
2. Normalize the pivot row: Divide the row by the pivot to make the pivot element 1.
3. Eliminate entries below the pivot: Use row operations to make all entries below the pivot in the column zero.
4. Move to the next column and row: Repeat the process for the next pivot (next diagonal element).
5. Continue until all rows or columns are exhausted, leaving an upper triangular form.

Example:

Consider the matrix:

1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$$ 1. Pivot in the first column: The pivot is $$1$$. Eliminate entries below: - $$R_2' = R_2 - 4R_1$$ - $$R_3' = R_3 - 7R_1$$ $$\begin{bmatrix} 1 & 2 & 3 \\ 0 & -3 & -6 \\ 0 & -6 & -12 \end{bmatrix}$$ 2. Pivot in the second column: The pivot is $$-3$$. Normalize $$R_2$$ and eliminate the entry below: - $$R_2' = R_2 / -3$$ - $$R_3' = R_3 - (-6)R_2$$ $$\begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{bmatrix}$$ 3. Result: The matrix is in upper triangular form (row echelon form). ### Rank The **rank** of a matrix is the number of non-zero rows in its row echelon form. Here, the rank of $$A$$ is **2** (two non-zero rows). --- Would you like detailed examples or have any questions? Here are five related questions to deepen understanding: 1. What is the difference between row echelon form and reduced row echelon form? 2. How do row operations affect the determinant of a matrix? 3. How do we determine the rank of a matrix from its echelon form? 4. What is the significance of rank in solving systems of linear equations? 5. How is rank related to the invertibility of a matrix? **Tip**: To avoid errors, carefully check each step of the row operations to maintain numerical accuracy, especially for large matrices.