Reading the Shape of a Rectangular Matrix
Row reduction works for rectangular matrices as well as square ones, but the shape affects which pivot patterns are possible. A tall matrix has more rows than columns, so an echelon form must have at least one zero row. A wide matrix has more columns than rows, so some columns cannot contain pivots. In either case, a pivot need not appear in the first column: if that column has no suitable nonzero entry, the search continues to the right.
For a matrix, a leading entry is the first nonzero entry in a nonzero row. In echelon form, all zero rows are below the nonzero rows, and the leading entries move strictly to the right as we move down the nonzero rows. A pivot position is the position of a leading entry; its column is a pivot column. Columns without pivot positions are nonpivot columns. In reduced row echelon form (RREF), each leading entry is \(1\), and each pivot is the only nonzero entry in its column.
Row operations are performed on entire rows, not on selected entries. The allowed operations are interchanging two rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another row. Each operation preserves the matrix dimensions. Each is also reversible: an interchange undoes itself, multiplication by \(c\ne0\) is undone by multiplying by \(1/c\), and adding \(c\) times one row to another is undone by adding \(-c\) times that row. Therefore, every displayed sequence below produces a matrix row-equivalent to the original one. Whether the result is echelon or reduced echelon is checked separately from its entries.
What the Dimensions Allow
Proof. Each pivot belongs to a nonzero row, and different nonzero rows have pivots in different columns because their leading entries move strictly right as we move down. Thus \(p\) is the number of nonzero rows and also the number of pivot columns. Of the \(m\) rows, the remaining \(m-p\) are zero; of the \(n\) columns, the remaining \(n-p\) have no pivots. Since \(p\le m\) and \(p\le n\), we have \(p\le\min(m,n)\). If \(m>n\), then \(m-p\ge m-n\), and if \(n>m\), then \(n-p\ge n-m\). \(\square\)
This count is about the shape of an echelon form; it does not say which columns contain pivots. That depends on the entries. In particular, do not assume that the first column is a pivot column, or that every column of a wide matrix is a pivot column. The row reduction itself determines the positions.
Proof. By the definition of RREF, a pivot is \(1\), and it is the only nonzero entry in its column. Therefore, the pivot column has a \(1\) at the pivot position and zeros in all other positions, which is exactly a standard unit column. \(\square\)
Worked Examples: Tall and Wide Matrices
Worked Example: A Tall Matrix with Zero Rows
Reduce the following \(5\times3\) matrix to RREF. This matrix has more rows than columns, so its echelon form must have at least two zero rows.
Use \(R_2\to R_2-2R_1\) and \(R_4\to R_4-R_1\). The rows become \(R_2=(0,0,1)\) and \(R_4=(0,0,3)\); the other rows are unchanged:
Now apply \(R_3\to R_3-R_2\) and \(R_4\to R_4-3R_2\). The third and fourth rows become zero:
This is echelon form, with pivots in columns 1 and 3. To make it reduced, apply \(R_1\to R_1-R_2\):
Check: The final matrix has leading \(1\)s in columns 1 and 3, and each pivot is the only nonzero entry in its column. Its three zero rows are at the bottom. The final first row is \(3(1,2,1)-(2,4,3)=(1,2,0)\), and the second row is \((2,4,3)-2(1,2,1)=(0,0,1)\). The original third row equals the final second row, and the original fourth row is \((1,2,1)+3(0,0,1)\). Thus the stated rows and zeros are consistent with the row operations.
Worked Example: A Wide Matrix with Skipped Columns
Reduce this \(3\times5\) matrix to RREF. Columns 1 and 3 will not contain pivots.
First apply \(R_2\to R_2-2R_1\). The new second row is \((0,0,0,1,1)\). Then apply \(R_3\to R_3-R_2\), using this new second row; the third row becomes \((0,0,0,0,1)\).
To clear the entries above the pivot in column 5, apply \(R_2\to R_2-R_3\) and \(R_1\to R_1-R_3\):
Check: The leading entries are in columns 2, 4, and 5, moving strictly right as the rows descend; the pivot columns contain no other nonzero entries. The first column remains zero, and column 3 has no pivot. For an additional row check, the final rows are \(\mathbf{b}_2-\mathbf{b}_1-\mathbf{b}_3=(0,1,2,0,0)\), \(2\mathbf{b}_2-4\mathbf{b}_1-\mathbf{b}_3=(0,0,0,1,0)\), and \(\mathbf{b}_3-\mathbf{b}_2+2\mathbf{b}_1=(0,0,0,0,1)\), where \(\mathbf{b}_1,\mathbf{b}_2,\mathbf{b}_3\) here denote the original rows of \(\mathbf{B}\).
Worked Example: Several Rows Become Zero
Reduce this \(4\times5\) matrix. Its nonzero rows are dependent in pairs, so the echelon form will have two zero rows.
Apply \(R_2\to R_2-2R_1\) and \(R_4\to R_4-2R_3\). Each of the changed rows becomes zero:
The zero row in position 2 must move below the nonzero rows. Apply \(R_2\leftrightarrow R_3\):
This matrix is already in reduced row echelon form, so no further row operation is needed: the leading \(1\) in column 3 is the only nonzero entry in that column, and the leading \(1\) in column 5 is the only nonzero entry in its column.
Check: The first row has leading entry \(1\) in column 3, and the second has leading entry \(1\) in column 5. These are the only nonzero entries in their respective columns, so the displayed matrix is RREF. The original second row was twice the first, and the original fourth row was twice the third; subtracting those multiples gives the two zero rows. The remaining nonzero rows are \((0,0,1,2,0)\) and \((0,0,0,0,1)\), as displayed.
A Reliable Procedure and a Common Pitfall
When reducing a rectangular matrix, the row count does not tell you where to place a pivot. Work from left to right through the columns, and from top to bottom through the rows. If the current column has no usable nonzero entry in the rows still under consideration, leave that column without a pivot and continue to the right. Once a pivot is chosen, clear the entries below it to obtain echelon form; for RREF, also clear entries above it and scale each pivot to \(1\). If a row becomes zero, move it below the nonzero rows before declaring the matrix echelon.
In the rows not yet assigned pivots, find a nonzero entry. A column with no such entry is skipped.
Interchange rows if needed to put a nonzero entry in the next pivot row. For RREF, scale that row so the pivot is \(1\).
Use row replacements to make the entries below the pivot zero for echelon form, and above it zero as well for RREF.
Confirm that pivots move strictly right, zero rows are at the bottom, and for RREF each pivot is the only nonzero entry in its column.
Common mistake: forcing a pivot into the first column. In the wide-matrix example, every entry in column 1 is zero. A mistaken calculation might try to make the first entry of row 1 a pivot by dividing \(R_1\) by \(a_{11}=0\). Division by zero is undefined, so this is not a valid row operation. The correction is to skip column 1 and use the first nonzero entry of row 1, which is \(1\) in column 2. The resulting pivot positions are columns 2, 4, and 5; a skipped column is not an error.
Another check is to count the rows and columns after reduction. In the \(5\times3\) example, two pivots leave three zero rows, as required by the shape bound. In the \(3\times5\) example, three pivots leave two nonpivot columns. These counts are useful checks, but they cannot replace checking the actual entries: only the row reduction determines which particular columns are pivot columns.
Check Your Understanding
For each question, use the echelon definitions and verify any row operations you propose.
- How many zero rows must an echelon form of a \(6\times4\) matrix have at minimum?
- How many nonpivot columns must an echelon form of a \(3\times7\) matrix have at minimum?
- For the wide matrix in the second example, list its pivot columns and its nonpivot columns.
- Why must zero rows be moved to the bottom before a matrix can be called echelon form?
- In RREF, what entries appear in a pivot column other than the pivot itself?
