Prove that the interchange of two rows of a matrix can be accomplished by a finite sequence of elementary row operations of the other two types.
Solution
Suppose the two rows to be interchanged are \(R_i\) and \(R_j\).
1. Replace \(R_i\) by \(R_i+R_j\).
This is an elementary operation of the type “add a scalar multiple of one row to another.” The pair becomes \[ (R_i+R_j,\ R_j). \]
2. Replace \(R_j\) by \(R_j-R_i\), where \(R_i\) now denotes the new first row.
The pair becomes \[ (R_i+R_j,\ -R_i). \]
3. Replace \(R_i\) by \(R_i+R_j\).
The pair becomes \[ (R_j,\ -R_i). \]
4. Multiply \(R_j\) by \(-1\).
This is an elementary row scaling, and the pair becomes \[ (R_j,R_i). \] Thus the original rows have been interchanged using only the other two kinds of elementary row operations.