16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 1.3.7
Systems of Linear Equations
Mathematics — Linear Algebra · elementary-row-operations
Problem
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.Intuitive explanation
The key insight is that row swaps can be built from row scaling and row addition—you don't need them as a primitive operation. Think of it like moving two objects past each other: you can shuffle one row "through" another by adding scaled versions back and forth, similar to how you'd swap two items in a line by moving one aside, sliding the other over, then moving the first into its new spot. Work out what sequence of additions and scalings achieves this, and you'll see why row interchanges are redundant.
Intuitive explanation
The key insight is that row swaps can be built from row scaling and row addition—you don't need them as a primitive operation. Think of it like moving two objects past each other: you can shuffle one row "through" another by adding scaled versions back and forth, similar to how you'd swap two items in a line by moving one aside, sliding the other over, then moving the first into its new spot. Work out what sequence of additions and scalings achieves this, and you'll see why row interchanges are redundant.