Elementary transformations

1. What an Elementary Transformation Is

An elementary transformation (elementary operation) is a simple, reversible operation performed on the rows or columns of a matrix. These operations do not change the essential information of the matrix — in particular they preserve its rank — and they are the machinery behind Gaussian elimination, finding inverses and solving linear systems.

2. The Six Elementary Operations

#	Row operation	Symbol	Column operation
1	Interchange of two rows	Rᵢ ↔ Rⱼ	Cᵢ ↔ Cⱼ
2	Multiplication of a row by a non-zero scalar	Rᵢ → k Rᵢ (k ≠ 0)	Cᵢ → k Cᵢ (k ≠ 0)
3	Addition of k times another row to a row	Rᵢ → Rᵢ + k Rⱼ	Cᵢ → Cᵢ + k Cⱼ

Note the restriction k ≠ 0 for operation 2 — multiplying by 0 would destroy information and is not an elementary operation.

3. Why Rank is Preserved

Each operation is the result of multiplying the matrix by a non-singular elementary matrix:

interchanging rows = multiplying by a permutation matrix;
multiplying a row by k = multiplying by a diagonal matrix with k in place of 1;
adding k times another row = multiplying by a lower/upper triangular matrix with unit diagonal.

All these elementary matrices have determinant 1 or k ≠ 0, so they are invertible. Since multiplication by an invertible matrix does not change rank, the transformed matrix has the same rank as the original. Two matrices related this way are called equivalent, written A ~ B, and equivalence is a transitive relation.

Warning: row operations preserve the row space and column operations preserve the column space, but a mixture of both changes the column space — which is exactly why the normal form method may be used for rank but never for solving a system of equations (solutions must be found with row operations only).

4. Effect on the Determinant

Operation	Effect on det(A)
Rᵢ ↔ Rⱼ	det changes sign
Rᵢ → k Rᵢ	det is multiplied by k
Rᵢ → Rᵢ + k Rⱼ	det unchanged

These rules make evaluation of large determinants fast: reduce to triangular form, then multiply the diagonal entries.

5. Three Standard Uses

(a) Solving a system by Gauss–Jordan elimination (row operations only)

   x + 2y +  z = 4
  2x +  y -  z = 1
   x -  y + 2z = 3

  [ 1   2   1 | 4 ]        R2 → R2 - 2R1      [ 1   2   1 |  4 ]
  [ 2   1  -1 | 1 ]   ───► R3 → R3 - R1   ──► [ 0  -3  -3 | -7 ]
  [ 1  -1   2 | 3 ]                            [ 0  -3   1 | -1 ]

                            R3 → R3 - R2      [ 1   2   1 |  4 ]
                                              [ 0  -3  -3 | -7 ]
                                              [ 0   0   4 |  6 ]
  Back substitution:  z = 6/4 = 1.5 ,  -3y = -7 + 3z ⇒ y = 7/3 - 1.5 = 0.8333 ,
  x = 4 - 2y - z = 4 - 1.6667 - 1.5 = 0.8333

(b) Finding an inverse: [A | I] → [I | A⁻¹] (row operations only)

  A = [ 2  1 ]      [A | I] = [ 2  1 | 1  0 ]   R1 → R1/2     [ 1  0.5 | 0.5  0 ]
      [ 5  3 ]                [ 5  3 | 0  1 ]   R2 → R2-5R1   [ 0  0.5 |-2.5  1 ]
                                                 R2 → 2R2      [ 0  1   | -5   2 ]
                                                 R1 → R1-0.5R2 [ 1  0   |  3  -1 ]

  A⁻¹ = [  3  -1 ]        Check: A·A⁻¹ = [ 2  1 ][ 3  -1 ] = [ 1  0 ]  ✓
        [ -5   2 ]                       [ 5  3 ][ -5  2 ]   [ 0  1 ]
  (For this A, det = 2×3 − 1×5 = 1, so no fractions appear beyond halves.)

(c) Reducing a matrix to echelon / normal form — the standard route to finding rank (next two topics).

6. Points That Carry Marks

There are three types of elementary operations, in two flavours (row and column) — six in total; the row ones are the ones used for systems.
Interchange and addition are always allowed; scaling is allowed only with k ≠ 0.
Elementary operations preserve rank, determinant up to a known factor, and invertibility — but a column operation changes a system's solution set.
The inverse of an elementary operation is again an elementary operation of the same type, which is why "elementary" transformations can always be undone.

7. Exam Points

Define elementary row/column transformations and list all six, with e.g. Rᵢ → Rᵢ + kRⱼ notation — 5 marks, almost certain.
Show that elementary transformations do not change the rank of a matrix — 5 marks.
Use row operations to find A⁻¹ for a given 2×2 or 3×3 matrix — 5-7 marks, very frequent.
Solve a 3×3 system by Gauss–Jordan reduction (show every row operation) — 5-7 marks, frequent.
State the effect of each elementary row operation on the determinant — 3-5 marks.
Why are column operations not allowed when solving a system of linear equations? — 3 marks.

Elementary transformations