Matrices
1.8 Rank of a Matrix
Rank of a matrix via row echelon form, full rank, the rank-nullity theorem, null space and homogeneous systems, and solvability of AX = B.
1.8 Rank of a Matrix
Definition 1.32: The rank of A, Rank(A), is the number of nonzero rows after reducing A to row echelon form. It equals the number of linearly independent rows.
Example 1.30: Find Rank(M) for M = [[−1,6,−2],[3,1,−1],[2,−10,3]]
| Step | Operation | Result |
|---|---|---|
| Start | — | [−1,6,−2 | 3,1,−1 | 2,−10,3] |
| 1 | R₂→R₂+3R₁, R₃→R₃+2R₁ | [1,−6,2 | 0,19,−7 | 0,2,−1] |
| 2 | R₃⇄R₂/2, further ops | [1,0,−1 | 0,1,−½ | 0,0,5/2] |
| 3 | Scale R₃ | [1,0,0 | 0,1,0 | 0,0,1] = I₃ |
All 3 rows nonzero → Rank(M) = 3 (full rank)
Example 1.31: M = [[−1,6,−6],[3,1,−20],[2,−10,8]] → reduces to [[−1,6,−6],[0,1,−2],[0,0,0]] → Rank = 2 (one zero row)
Fundamental Theorem
Theorem 1.3: Rank(A) = Rank(AT) and Rank(A) ≤ min(m,n).
If Rank(A) = min(m,n), then A has full rank.
If Rank(A) = min(m,n), then A has full rank.
1.9 Null Space
Definition 1.34: The null space (kernel) of A is:
Null(A) = {X | AX = 0}
The nullity = dimension of the null space = number of free parameters in the solution.
Null(A) = {X | AX = 0}
The nullity = dimension of the null space = number of free parameters in the solution.
Rank-Nullity Theorem
nullity(A) + Rank(A) = n (number of columns)
Example — full rank: For M from Ex 1.30 (Rank = 3, n = 3):
AX = 0 → X = 0 (trivial only). nullity = 0. Check: 0 + 3 = 3 ✓
Example — rank 2: For M from Ex 1.31 (Rank = 2, n = 3):
Reduced: [−1,6,−6 | 0,1,−2 | 0,0,0] → x = 6t, y = 2t, z = t
| 6t |
| 2t |
| t |
Check: nullity(1) + Rank(2) = 3 = n ✓
Properties of AX = 0 (Homogeneous)
- Always consistent (X = 0 is always a solution)
- Either unique solution X = 0, or infinitely many
- If A is n×n: unique solution iff det(A) ≠ 0
- If X₁ and X₂ are solutions, so is αX₁ + βX₂
1.10 Solvability of AX = B
The inhomogeneous system AX = B is consistent iff Rank(A) = Rank([A|B]).
The general solution = particular solution Y + any solution X of AX = 0.
The general solution = particular solution Y + any solution X of AX = 0.