Elementary Matrices for the Three Row Operations

From a Row Operation to a Matrix

An elementary matrix is built by making one permitted row operation on an identity matrix. This turns a description such as “swap rows 1 and 3” into a specific matrix with a specific order. The construction is direct: first write the identity matrix, then change exactly the row or rows named by the operation. The identity matrix must have enough rows for every index used.

All matrices and scalars in this tutorial are over \(\mathbb{R}\). For \(n\in\mathbb{N}\), the identity matrix \(\mathbf{I}_n\) has order \(n\), and its \(k\)th row is \(\mathbf{e}_k^{\top}\), where \(\mathbf{e}_k\) is the \(k\)th standard basis column in \(\mathbb{R}^n\). Thus, for example, the rows of \(\mathbf{I}_3\) are \((1,0,0)\), \((0,1,0)\), and \((0,0,1)\).

The three elementary row operations are: swapping two distinct rows \(R_i\) and \(R_j\); scaling a row \(R_i\) by a nonzero scalar \(c\); and replacing one row \(R_i\) by \(R_i+cR_j\), where \(i\ne j\) and \(c\in\mathbb{R}\). The indices name rows, not the number of rows in the matrix. Consequently, the order \(n\) must be at least as large as the greatest row index named by the operation. For instance, an operation involving row 3 requires an identity matrix of order at least 3.

The Three Row Patterns

The definition gives a simple way to describe every row of an elementary matrix. A row not named by the operation stays equal to its row in the identity matrix. A named row is changed according to the operation. In the swap case, both named rows change places.

Proof. The \(k\)th row of \(\mathbf{I}_n\) is \(\mathbf{e}_k^{\top}\). A swap interchanges the original rows \(i\) and \(j\), so those rows become \(\mathbf{e}_j^{\top}\) and \(\mathbf{e}_i^{\top}\), respectively; all other rows are untouched. Scaling row \(i\) multiplies its original row \(\mathbf{e}_i^{\top}\) by \(c\), while leaving every other row untouched. For a replacement, the original row \(i\) is \(\mathbf{e}_i^{\top}\) and the original row \(j\) is \(\mathbf{e}_j^{\top}\), so the new row \(i\) is \(\mathbf{e}_i^{\top}+c\mathbf{e}_j^{\top}\); the other rows are untouched. These descriptions specify every row, so in each case they determine exactly one matrix. \(\square\)

The replacement condition \(i\ne j\) matters: the row being changed and the row being added must be different. The scaling condition \(c\ne0\) also matters. Multiplying a row by zero is not one of the permitted elementary row operations.

Proof. In a replacement, row \(i\) is \(\mathbf{e}_i^{\top}+c\mathbf{e}_j^{\top}\). Since \(i\ne j\), the \(i\)th component of this row is \(1+0=1\), and its \(j\)th component is \(0+c=c\). Every other component is zero, and every other row remains an identity row. In a scaling, row \(i\) is \(c\mathbf{e}_i^{\top}\), whose \(i\)th component is \(c\) and whose other components are zero; all other rows remain unchanged. This proves both entry descriptions. \(\square\)

Worked Constructions

Common Mistakes and Practical Checks

Using an identity matrix with too few rows. A tempting but invalid construction for \(R_1\leftrightarrow R_3\) is to start with \(\mathbf{I}_2\) and attempt to swap its first and third rows. The matrix \(\mathbf{I}_2\) has only rows 1 and 2, so row 3 does not exist and the operation cannot be performed. The correction is to use \(\mathbf{I}_3\), since the order must be at least the greatest named row index, 3. The corrected result is \(\begin{bmatrix}0&0&1\\0&1&0\\1&0&0\end{bmatrix}\).

Changing the wrong row in a replacement. For \(R_1\to R_1+3R_2\), changing row 2 instead would produce \(\begin{bmatrix}1&0\\3&1\end{bmatrix}\), which corresponds to \(R_2\to R_2+3R_1\), not the requested operation. The replacement changes the row on the left of the arrow; the row on the right is used as the added row and stays unchanged.

Allowing a zero scaling factor. Replacing \(R_i\to cR_i\) with \(c=0\) would erase row \(i\). For example, scaling row 2 of \(\mathbf{I}_2\) by zero gives \(\begin{bmatrix}1&0\\0&0\end{bmatrix}\), but this is not an elementary matrix for a permitted scaling operation: the scaling factor must be nonzero.

Before accepting a construction, check the order, the changed row or rows, and the entries in those rows. A swap must exchange exactly two identity rows; a scaling must alter exactly the specified row; and a replacement must alter the target row by adding the stated multiple of a different row. These checks make the construction mechanical and help prevent index errors.