Consistency of a linear system

A system AX = B is consistent if rank A = rank [A|B].

[A|B] = [1 −6 −1 | 10]
        [2 −2  3 | 10]
        [3 −8  2 | 20]

Step 1. R₂ → R₂ − 2R₁, R₃ → R₃ − 3R₁:

[1  −6  −1 |  10]
[0  10   5 | −10]
[0  10   5 | −10]

Step 2. R₃ → R₃ − R₂:

[1  −6  −1 |  10]
[0  10   5 | −10]
[0   0   0 |   0]

Consistency

rank A = 2 = rank [A|B]. The system is consistent. As the rank (2) is less than the number of unknowns (3), it has infinitely many solutions depending on one parameter.

Solution

Let z = k.

• 10y + 5k = −10 ⇒ y = −1 − k/2
• x − 6y − z = 10 ⇒ x = 10 + 6(−1 − k/2) + k = 4 − 2k

Answer: x = 4 − 2k, y = −1 − k/2, z = k, for any real k.

Check with k = 2: (0, −2, 2): 0 + 12 − 2 = 10 ✓; 0 + 4 + 6 = 10 ✓; 0 + 16 + 4 = 20 ✓.