Elementary row operation is matrix pre-multiplication

Dependencies:

1. Field
2. Every elementary row operation has a unique inverse
3. Identity matrix
4. Comparing coefficients of a polynomial with disjoint variables
5. Inverse of a matrix

Let $A$ be a $m$ by $n$ matrix over a field. Let $f$ be an elementary row operation. Then there exists a unique $m$ by $m$ matrix $R$ such that $f(A) = RA$. $R$ is also invertible.

By plugging in $A = I_m$ (identity matrix), we get $R = f(I_m)$. Therefore, the maxtrix associated with a row operation is the one obtained by applying that operation on the identity matrix.

Proof

We will show that there is a unique matrix $R$ which satisfies $f(A) = RA$ for all $A$.

• $\langle p \rangle \leftarrow c\langle p \rangle$: \[ (RA)[i, j] = \sum_k R[i, k] A[k, j] = \begin{Bmatrix} c & i = p \\ 1 & i \neq p \end{Bmatrix}A[i, j] \] On comparing coefficients of $A[*, j]$, we get \[ R[i, j] = \begin{cases} c & i = j = p \\ 1 & j = i \neq p \\ 0 & i \neq j \end{cases} \]

• $\langle p \rangle \leftarrow \langle p \rangle + c\langle q \rangle$: \[ (RA)[i, j] = \sum_k R[i, k] A[k, j] = \begin{cases} A[i, j] & i \neq p \\ A[p, j] + cA[q, j] & i = p \end{cases} \] On comparing coefficients of $A[*, j]$, we get \[ R[i, j] = \begin{cases} 0 & i \neq p \wedge i \neq j \\ 1 & i \neq p \wedge i = j \\ 0 & i = p \wedge j \neq p \wedge j \neq q \\ 1 & i = p \wedge j = p \\ c & i = p \wedge j = q \end{cases} \]

• $\langle p \rangle \leftrightarrow \langle q \rangle$: \[ (RA)[i, j] = \sum_k R[i, k] A[k, j] = \begin{cases} A[i, j] & i \neq p \wedge i \neq q \\ A[q, j] & i = p \\ A[p, j] & i = q \end{cases} \] On comparing coefficients of $A[*, j]$, we get \[ R[i, j] = \begin{cases} 0 & i \neq p \wedge i \neq q \wedge i \neq j \\ 1 & i \neq p \wedge i \neq q \wedge i = j \\ 0 & i = p \wedge j \neq q \\ 1 & i = p \wedge j = q \\ 0 & i = q \wedge j \neq p \\ 1 & i = q \wedge j = p \end{cases} \]

• Invertibility:

  We know that every elementary row operation has a unique inverse. Let $R$ be the matrix corresponding to an elementary row operation and let $S$ be the matrix of its inverse operation.

  • Since $S$ represents the inverse operation of $R$, $S(RA) = A$ for all $A$. For $A = I_m$, we get $SR = I_m$.
  • Since $R$ represents the inverse operation of $S$, $R(SA) = A$ for all $A$. For $A = I_m$, we get $RS = I_m$.

  Therefore, $R$ is invertible and its inverse is $S$.

Dependency for:

1. Row equivalence matrix
2. Determinant of product is product of determinants

Info:

• Depth: 5
• Number of transitive dependencies: 13

Transitive dependencies:

1. Group
2. Ring
3. Polynomial
4. Integral Domain
5. Comparing coefficients of a polynomial with disjoint variables
6. Field
7. Semiring
8. Matrix
9. Matrix multiplication is associative
10. Elementary row operation
11. Every elementary row operation has a unique inverse
12. Identity matrix
13. Inverse of a matrix