From Row Operations to Row Equivalence
One elementary row operation changes a matrix in a controlled way: it swaps two rows, multiplies a row by a nonzero scalar, or adds a multiple of one row to a different row. A calculation may use several such operations in succession. Row equivalence gives a name to the relationship between the starting matrix and the matrix at the end. It lets us discuss the transformation as a whole without confusing it with equality of the two matrices.
Throughout this tutorial, matrices have real entries. Each elementary row operation acts on the rows of a matrix: it does not change the number of rows or columns. The scalar used to multiply a row must be nonzero; otherwise the operation could erase information from that row. In an operation that adds a multiple of one row to another, the source and target rows are distinct.
The phrase “finite sequence” matters: a single operation is allowed, several operations are allowed, and the empty sequence is allowed. In a written chain, every step must be justified by one elementary row operation. For example, the notation \(\mathbf{A}\sim\mathbf{B}\) asserts that there is such a sequence; it does not assert that corresponding entries of \(\mathbf{A}\) and \(\mathbf{B}\) are equal.
Reading and Recording an Operation Sequence
Worked Example: A Swap Followed by a Row Addition
Start with \(\mathbf{A}=\begin{bmatrix}1&2&0\\3&1&4\end{bmatrix}\). Swap the two rows, then replace the new second row by itself minus one third of the first row. Record the sequence and its endpoint.
First apply \(R_1\leftrightarrow R_2\). The result is
Now apply \(R_2\to R_2-\tfrac13R_1\). Entry by entry, the new second row is \((1,2,0)-\tfrac13(3,1,4)=(0,\tfrac53,-\tfrac43)\). Thus the full chain is
Check: The final first row is the original second row, as required by the swap. The final second row is \((1-\tfrac13\cdot3,\;2-\tfrac13\cdot1,\;0-\tfrac13\cdot4)=(0,\tfrac53,-\tfrac43)\). Each arrow records one permitted operation, so the endpoint is row-equivalent to the starting matrix.
For an operation to be legal, its scalar and row indices must meet the conditions in the definition. In particular, scaling a row by \(0\) is not an elementary row operation. Likewise, the row-addition operation uses one row to change a different row; the notation \(R_i\to R_i+cR_j\) requires \(i\ne j\).
Worked Example: Several Operations on an Augmented Matrix
Apply \(R_2\to R_2-2R_1\), then \(R_2\to-\tfrac13R_2\), then \(R_1\to R_1-2R_2\) to the augmented matrix \(\begin{bmatrix}1&2&5\\2&1&4\end{bmatrix}\).
After the first operation, the second row is \((2,1,4)-2(1,2,5)=(0,-3,-6)\):
Multiplying the second row by \(-\tfrac13\) gives \((0,1,2)\). Then subtracting twice that row from the first gives \((1,2,5)-2(0,1,2)=(1,0,1)\). The complete chain is
Check: The new second row after scaling is \(-\tfrac13(0,-3,-6)=(0,1,2)\). The new first row is \((1,2,5)-2(0,1,2)=(1,0,1)\). All three steps are elementary row operations, so the displayed matrices are row-equivalent in succession. When the matrix represents an augmented linear system, the earlier result on why row operations preserve solutions says that each step preserves its solution set.
Row Equivalence Is an Equivalence Relation
A relation is an equivalence relation if it is reflexive, symmetric, and transitive. For row equivalence, these properties follow directly from how finite operation sequences work. Symmetry requires an additional observation: every elementary row operation can be undone by another elementary row operation.
Proof. To prove reflexivity, take any matrix \(\mathbf{A}\). The empty sequence of operations leaves it unchanged, so \(\mathbf{A}\sim\mathbf{A}\).
For symmetry, suppose \(\mathbf{A}\sim\mathbf{B}\). Then some finite sequence of elementary row operations transforms \(\mathbf{A}\) into \(\mathbf{B}\). Each operation has an elementary inverse: a row swap is undone by the same swap; multiplying a row by a nonzero scalar \(c\) is undone by multiplying that row by \(1/c\); and \(R_i\to R_i+cR_j\) is undone by \(R_i\to R_i-cR_j\). Apply these inverse operations in reverse order to transform \(\mathbf{B}\) back into \(\mathbf{A}\). Therefore \(\mathbf{B}\sim\mathbf{A}\).
For transitivity, suppose \(\mathbf{A}\sim\mathbf{B}\) and \(\mathbf{B}\sim\mathbf{C}\). The first relation gives a finite sequence from \(\mathbf{A}\) to \(\mathbf{B}\), and the second gives a finite sequence from \(\mathbf{B}\) to \(\mathbf{C}\). Performing the first sequence and then the second gives a finite sequence from \(\mathbf{A}\) to \(\mathbf{C}\). Therefore \(\mathbf{A}\sim\mathbf{C}\). All three properties hold, so row equivalence is an equivalence relation. \(\square\)
The proof explains a practical feature of operation chains: a chain can be reversed. To reverse it, undo its last operation first, then its preceding operation, and continue until the starting matrix is reached. This is also why the notation \(\mathbf{A}\sim\mathbf{B}\) describes a symmetric relationship, even though a particular written calculation is usually read from left to right.
Equivalence Is Not Equality
Equality of matrices means that they have the same size and every entry in one matrix equals the corresponding entry in the other. Row equivalence is different: it means that a permitted sequence connects the matrices. A row operation can change entries, so equal matrices are row-equivalent, but row-equivalent matrices need not be equal.
Worked Example: Different Matrices That Are Row-Equivalent
Compare \(\mathbf{A}=\begin{bmatrix}1&2\\3&4\end{bmatrix}\) and \(\mathbf{B}=\begin{bmatrix}1&2\\0&-2\end{bmatrix}\). The matrices are not equal because their lower-left entries differ. Check whether they are row-equivalent.
Apply \(R_2\to R_2-3R_1\) to \(\mathbf{A}\):
The result is exactly \(\mathbf{B}\), since \((3,4)-3(1,2)=(0,-2)\). Thus \(\mathbf{A}\sim\mathbf{B}\), while \(\mathbf{A}\ne\mathbf{B}\). The calculation checks both claims: one entry differs, and a single legal operation connects the matrices.
Row equivalence is defined only for matrices of the same size, since elementary row operations do not change the number of rows or columns. Even among matrices of the same size, not every pair is row-equivalent.
Proof. Every elementary row operation applied to a zero matrix leaves it a zero matrix: swapping zero rows changes nothing, multiplying a zero row by a scalar leaves it zero, and adding a multiple of a zero row to another zero row still gives a zero row. Therefore every finite sequence starting from a zero matrix ends at a zero matrix. If a matrix \(\mathbf{A}\) is row-equivalent to the zero matrix, symmetry of row equivalence gives a sequence from the zero matrix to \(\mathbf{A}\). The preceding observation forces \(\mathbf{A}\) to be the zero matrix. Conversely, the zero matrix is row-equivalent to itself by the empty sequence. \(\square\)
Worked Example: A Pair That Is Not Row-Equivalent
Let \(\mathbf{O}=\begin{bmatrix}0&0\\0&0\end{bmatrix}\) and \(\mathbf{C}=\begin{bmatrix}1&0\\0&0\end{bmatrix}\). Both are \(2\times2\) matrices, but they are not row-equivalent.
The matrix \(\mathbf{C}\) is not the zero matrix, because its upper-left entry is \(1\). By the theorem, the zero matrix can be row-equivalent only to itself. Hence \(\mathbf{O}\not\sim\mathbf{C}\).
Check: Any sequence of row operations that starts at \(\mathbf{O}\) leaves every entry zero, so it cannot produce the entry \(1\) in \(\mathbf{C}\). In the reverse direction, if a sequence carried \(\mathbf{C}\) to \(\mathbf{O}\), symmetry would give a sequence from \(\mathbf{O}\) to \(\mathbf{C}\), which is impossible for the same reason.
Common Mistake: Treating an Arrow as Equality
A common error is to write that a row operation makes two matrices “equal.” For example, after \(R_2\to R_2-R_1\), the incorrect statement \(\begin{bmatrix}1&2\\3&4\end{bmatrix}=\begin{bmatrix}1&2\\2&2\end{bmatrix}\) claims that the lower rows already agree. They do not: the original lower row is \((3,4)\), while the new lower row is \((2,2)\).
The correction is to use row equivalence for the relationship between the starting and resulting matrices:
The operation is valid because \((3,4)-(1,2)=(2,2)\). The matrices remain different; the symbol \(\sim\) records that one can be obtained from the other by an elementary row operation.
When reading or writing a chain, check three things: each matrix has the same dimensions, each arrow names or clearly corresponds to a permitted row operation, and each resulting row has been calculated correctly. For augmented matrices, keep every coefficient and the right-hand-side entry in its row together when performing the operation.
Check Your Understanding
Use the definition and the operation rules to answer the following.
- Are \(\begin{bmatrix}1&0\\2&1\end{bmatrix}\) and \(\begin{bmatrix}1&0\\0&1\end{bmatrix}\) row-equivalent? Give the operation that decides the question.
- Which elementary operation undoes \(R_1\to -4R_1\)? State the operation and its scalar.
- Explain why the empty sequence proves that every matrix is row-equivalent to itself.
- Can a \(2\times3\) matrix be row-equivalent to a \(3\times2\) matrix? Explain using the definition.
- Prove that if \(\mathbf{A}\sim\mathbf{B}\), then \(\mathbf{B}\sim\mathbf{A}\), identifying how each of the three elementary operations is reversed.
