Rank of a matrix by reduction [6,1,3,8;4,2,6,−1;…]

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

A = [ 6 1  3  8]
    [ 4 2  6 −1]
    [10 3  9  7]
    [16 4 12 15]

Step 1. Eliminate the first column using R₁ (pivot 6): R₂ → 3R₂ − 2R₁, R₃ → 3R₃ − 5R₁, R₄ → 3R₄ − 8R₁:

R₂: (12−12, 6−2, 18−6, −3−16)  = (0, 4, 12, −19)
R₃: (30−30, 9−5, 27−15, 21−40) = (0, 4, 12, −19)
R₄: (48−48, 12−8, 36−24, 45−64)= (0, 4, 12, −19)

[6 1  3   8]
[0 4 12 −19]
[0 4 12 −19]
[0 4 12 −19]

Step 2. R₃ → R₃ − R₂, R₄ → R₄ − R₂:

[6 1  3   8]
[0 4 12 −19]
[0 0  0   0]
[0 0  0   0]

The matrix is in triangular (echelon) form with two non-zero rows.

Answer: rank(A) = 2.

(Check: in the original matrix R₃ = R₁ + R₂ and R₄ = R₁ + R₃, so only R₁ and R₂ are independent.)