Consistent and inconsistent systems

A system of equations is consistent if it has at least one solution and inconsistent if it has none.

The system a₁x + b₁y + c₁z = d₁, a₂x + b₂y + c₂z = d₂, a₃x + b₃y + c₃z = d₃ is the single matrix equation AX = B, with A the coefficient matrix, X the column (x, y, z) and B the column (d₁, d₂, d₃).

If |A| ≠ 0, the system has exactly one solution.

If |A| = 0, compute (adj A)B. If (adj A)B ≠ O, there is no solution and the system is inconsistent.

If |A| = 0 and (adj A)B = O, the test is not decisive: the system may have infinitely many solutions or none.

In two variables the picture is two lines: |A| ≠ 0 means the lines cross at one point; |A| = 0 means they are parallel (no solution) or the same line (infinitely many).

Example: x + 2y = 4 and 2x + 4y = 10 have |A| = 4 − 4 = 0 and adj A = [[4, −2], [−2, 1]] times (4, 10) gives (adj A)B = (−4, 2) ≠ O, so the system is inconsistent: the lines are parallel.