Exchange of Columns as Sequence of Other Elementary Column Operations

Theorem
Let $\mathbf A$ be an $m \times n$ matrix.

Let $i, j \in \closedint 1 m: i \ne j$

Let $\kappa_k$ denote the $k$th column of $\mathbf A$ for $1 \le k \le n$:
 * $\kappa_k = \begin {pmatrix} a_{1 k} \\ a_{2 k} \\ \vdots \\ a_{m k} \end {pmatrix}$

Let $e$ be the elementary column operation acting on $\mathbf A$ as:

Then $e$ can be expressed as a finite sequence of exactly $4$ instances of the other two elementary column operations.

Proof
In the below:
 * $\kappa_i$ denotes the initial state of column $i$
 * $\kappa_j$ denotes the initial state of column $j$


 * $\kappa_i'$ denotes the state of column $i$ after having had the latest elementary column operation applied
 * $\kappa_j'$ denotes the state of column $j$ after having had the latest elementary column operation applied.

$(1)$: Apply $\text {ECO} 2$ to column $j$ for $\lambda = 1$:


 * $\kappa_j \to \kappa_j + \kappa_i$

After this operation:

$(2)$: Apply $\text {ECO} 2$ to column $i$ for $\lambda = -1$:


 * $\kappa_i \to \kappa_i + \paren {-\kappa_j}$

After this operation:

$(3)$: Apply $\text {ECO} 2$ to column $j$ for $\lambda = 1$:


 * $\kappa_j \to \kappa_j + \kappa_i$

After this operation:

$(4)$: Apply $\text {ECO} 1$ to column $i$ for $\lambda = -1$:


 * $\kappa_i \to -\kappa_i$

After this operation:

Thus, after all the $4$ elementary column operations have been applied, we have:

Hence the result.

Also see

 * Exchange of Rows as Sequence of Other Elementary Row Operations