Rank of a matrix by reduction [1,3,−2,1;1,1,1,1;…]

The rank is the number of non-zero rows left when the matrix is reduced to echelon (and then normal) form.

A = [1  3 −2 1]
    [1  1  1 1]
    [2  0 −3 2]
    [3  3 −3 3]

Step 1. R₂ → R₂ − R₁, R₃ → R₃ − 2R₁, R₄ → R₄ − 3R₁:

[1  3 −2 1]
[0 −2  3 0]
[0 −6  1 0]
[0 −6  3 0]

Step 2. R₃ → R₃ − 3R₂, R₄ → R₄ − 3R₂:

[1  3 −2 1]
[0 −2  3 0]
[0  0 −8 0]
[0  0 −6 0]

Step 3. R₄ → R₄ − (3/4)R₃:

[1  3 −2 1]
[0 −2  3 0]
[0  0 −8 0]
[0  0  0 0]

Step 4 (to normal form). Use column operations. C₂ → C₂ − 3C₁, C₃ → C₃ + 2C₁ and C₄ → C₄ − C₁ clear the first row, leaving the rows (1, 0, 0, 0), (0, −2, 3, 0), (0, 0, −8, 0), (0, 0, 0, 0). Then C₃ → C₃ + (3/2)C₂ clears the 3 in the second row. Finally, R₂ → (−1/2)R₂ and R₃ → (−1/8)R₃ make the pivots 1, and the matrix becomes

[I₃ 0]
[0  0]

The normal form contains the 3 × 3 identity matrix, so

Answer: rank(A) = 3.