Rank of a matrix by reduction [2,−2,0,6;4,2,0,2;…]

The rank of a matrix is the order of its largest square sub-matrix with non-zero determinant. Equivalently, it is the number of non-zero rows in its row echelon form.

A = [2 −2 0 6]
    [4  2 0 2]
    [1 −1 0 3]
    [1 −2 1 2]

Step 1. R₁ ↔ R₃ to get a leading 1:

[1 −1 0  3]
[4  2 0  2]
[2 −2 0  6]
[1 −2 1  2]

Step 2. R₂ → R₂ − 4R₁, R₃ → R₃ − 2R₁, R₄ → R₄ − R₁:

[1 −1 0   3]
[0  6 0 −10]
[0  0 0   0]
[0 −1 1  −1]

Step 3. Interchange R₂ and R₄ (and move the zero row to the bottom):

[1 −1 0   3]
[0 −1 1  −1]
[0  6 0 −10]
[0  0 0   0]

Step 4. R₃ → R₃ + 6R₂:

[1 −1 0   3]
[0 −1 1  −1]
[0  0 6 −16]
[0  0 0   0]

This is in echelon form with 3 non-zero rows.

Answer: rank(A) = 3.

(Check: in the original matrix R₁ = 2R₃, which explains the zero row, and the three pivots show rank = 3.)