Rank of Matrix

Rank of Matrix

1. Definition

The rank of a matrix A, written ρ(A) or rank(A), is the maximum number of linearly independent rows (or columns) of A. Equivalently, and most usefully in problems:

rank = the order of the largest non-zero minor of A
rank = the number of non-zero rows left after reducing A to echelon form

All three definitions agree. (A minor of order r is the determinant of any r × r sub-matrix obtained by deleting rows and columns.)

2. Illustration of the Three Meanings

        [ 1   2   3 ]
   A =  [ 2   4   6 ]      R2 = 2R1,  R3 = 3R1
        [ 3   6   9 ]

   • independent rows       : only R1 ⇒ rank = 1
   • largest non-zero minor : order 1  (every 2×2 minor = 0, e.g. 1·4 − 2·2 = 0)
   • echelon form           : [ 1  2  3 ; 0  0  0 ; 0  0  0 ]  ⇒ 1 non-zero row

3. Properties of Rank (each is a possible short question)

1. ρ(A) ≤ min(m, n) for an m × n matrix — rank can never exceed the smaller dimension.
2. ρ(A) = 0 if and only if A is the null matrix.
3. ρ(A) = n for a square n × n matrix if and only if A is non-singular (det A ≠ 0). A singular n × n matrix has rank ≤ n − 1.
4. ρ(A) = ρ(Aᵀ) — row rank always equals column rank.
5. ρ(AᵀA) = ρ(AAᵀ) = ρ(A).
6. ρ(AB) ≤ min{ρ(A), ρ(B)} — multiplying can only reduce rank.
7. Rank is not changed by elementary transformations, nor by multiplying by a non-singular matrix.
8. Rank–nullity theorem: for an m × n matrix, ρ(A) + nullity(A) = n, where nullity is the dimension of the null space (the number of free variables in Ax = 0).
9. If A is n × n non-singular then ρ(A) = ρ(AB) = ρ(BA) = n for every B of compatible size.

4. Three Standard Methods

Method	How	When to use
Minor method	Find the highest order r for which some r × r minor ≠ 0; all minors of order r + 1 vanish	Small matrices, or when a parameter must satisfy the largest non-vanishing minor
Echelon form	Row-reduce (row operations only) and count the non-zero rows	Always — the standard, fastest method
Normal form	Reduce by row and column operations to [Iᵣ 0 ; 0 0]; then rank = r	When the question asks for the normal form as well

        [ 1   2  -1 ]
   A =  [ 2  -1   3 ]        det A = 1[(-1)(2) - (3)(1)] - 2[(2)(2) - (3)(3)] + (-1)[(2)(1) - (-1)(3)]
        [ 3   1   2 ]               = 1(-2 - 3) - 2(4 - 9) + (-1)(2 + 3)
                                     = -5 + 10 - 5 = 0     ⇒  rank < 3

   A 2×2 minor:  | 1   2 |  = (1)(-1) - (2)(2) = -1 - 4 = -5 ≠ 0
                 | 2  -1 |
   ⇒  a non-zero minor of order 2 exists, and no 3×3 minor is non-zero
   ⇒  ρ(A) = 2

This "split into cases on the parameter" pattern is the most common rank question in exams.

7. Worked Example 3 — Rank of a 4 × 3 Matrix

        [ 1   2   3 ]
        [ 2   4   6 ]                      R2-2R1, R3-3R1, R4-R1
   A =  [ 3   6   9 ]   ──────────►  rank = 1  (since rows 2,3,4 all vanish)

   But for   B = [ 1  2  3 ]            R2 → R2 - 2R1 = [ 0  0  -5 ]
                [ 2  2  1 ]             R3 → R3 - 3R1 = [ 0  0  -8 ]
                [ 3  2 -1 ]             then R3 → R3 - (8/5)R2 = [ 0  0   0 ]
                                        the remaining rows [1 2 3] , [0 0 -5] are independent
   ⇒ ρ(B) = 2                    (note ρ ≤ min(3, 3) and det B = 0)

8. Exam Points

• Define the rank of a matrix and give the three equivalent meanings — 5 marks, almost certain.
• State and use any five properties of rank (rank ≤ min(m,n), rank of null matrix, relation with singularity, ρ(A) = ρ(Aᵀ)) — 5 marks.
• Find the rank of a given matrix by the echelon method — 5-7 marks, very frequent.
• Find the rank by finding the largest non-zero minor — 5 marks.
• Find the values of k for which the rank of a matrix is 2 (or 3) — 5-7 marks, very frequent.
• State the rank–nullity theorem, ρ(A) + nullity(A) = n — 3 marks.
• Prove ρ(AB) ≤ min{ρ(A), ρ(B)} — 3-5 marks.