Theory
You already do this with equations
Solve these in your head: x + y = 5 and 2x + 2y = 10.
You probably halved the second equation, saw it was the first one again, and moved on. Halving an equation, swapping which one you write first, adding one to another: you juggle equations like this all the time, and the answer never changes.
Matrices allow exactly the same juggling. The moves have names, and exams love them.
Theory
Shuffling a marksheet
Take a class marksheet. Swap two students' rows: same data, new order. Convert one subject's marks from out-of-50 to out-of-100 by doubling a column: same performance, new scale. Nothing true about the class changed. Transformations rearrange a matrix without destroying what it says.
Theory
The three legal moves
For rows (columns work the same, with C instead of R):
- Swap: Rᵢ ↔ Rⱼ, two rows trade places.
- Scale: Rᵢ → kRᵢ, multiply every entry of one row by a constant k, and k must not be 0.
- Add a multiple: Rᵢ → Rᵢ + kRⱼ, add k times another row to this row.
That is the complete list. Any other move is illegal.
At a glance
The only three legal moves
| Move | Row form | Column form |
|---|---|---|
| Swap two lines | R₁ ↔ R₂ | C₁ ↔ C₂ |
| Scale a line by k (k ≠ 0) | R₁ → 3R₁ | C₁ → 3C₁ |
| Add a multiple of another | R₂ → R₂ + 2R₁ | C₂ → C₂ + 2C₁ |
Theory
Worked example: aim for zeros
The whole game is usually making zeros. Take A:
A = ⎡ 2 4 ⎤
⎣ 1 3 ⎦
Step 1: R₁ ↔ R₂, put the 1 on top (small leading numbers make the next step clean):
⎡ 1 3 ⎤
⎣ 2 4 ⎦
Step 2: R₂ → R₂ - 2R₁, to kill the 2 underneath: 2 - 2·1 = 0 and 4 - 2·3 = -2:
⎡ 1 3 ⎤
⎣ 0 -2 ⎦
A zero created, exactly where we wanted it.
Think first
Your turn
B = ⎡ 1 2 ⎤
⎣ 3 8 ⎦
Apply R₂ → R₂ - 3R₁ in your head. What does B become?
Show the answer
Work entry by entry on row 2: 3 - 3·1 = 0 and 8 - 3·2 = 2.
⎡ 1 2 ⎤
⎣ 0 2 ⎦
Row 1 does not change: the move rewrites only the row named on the left of the arrow.
Quiz
Apply C₁ ↔ C₂ to the matrix with rows (5 7) and (6 9). What do you get?
- Rows (7 5) and (9 6)
- Rows (6 9) and (5 7)
- Rows (5 6) and (7 9)
- Rows (9 7) and (6 5)
Show the answer
Rows (7 5) and (9 6)
Columns stand up and down, so C₁ ↔ C₂ swaps the two vertical lines: (5,6) trades places with (7,9), giving rows (7 5) and (9 6). Option B is a ROW swap, and option C is the transpose, both classic mix-ups.
Watch out
Where marks leak
Three habits that cost marks: transforming only some entries of a row (the move applies to every entry), scaling by k = 0 (it erases a whole row of information, so it is not allowed), and changing two rows in one step (R₂ → R₂ + kR₁ rewrites row 2 only, row 1 stays put). Write the move's name beside each step, examiners look for it.
Theory
Why these three moves matter
These moves are Gauss elimination, the method computers use to solve systems with thousands of equations. Your next topic, computing the inverse, can be done entirely with row transformations. And the "make zeros" instinct you just practised is the heart of both.
Summary
Key takeaways
- Row and column transformations rearrange a matrix without destroying its information.
- Only three moves are legal: swap (Rᵢ ↔ Rⱼ), scale by nonzero k (Rᵢ → kRᵢ), add a multiple (Rᵢ → Rᵢ + kRⱼ).
- Columns use the same moves with C notation.
- Each move rewrites one line only, and always every entry of it.
- The usual goal is creating zeros, the engine of Gauss elimination and inverses.
- Memory hook: swap, scale, add a multiple, nothing else.