A singular matrix in a network model
A small network of three nodes has flows x, y, z (in litres per minute) governed by the system Mx = b, where M is the 3×3 coefficient matrix with rows (1, 2, 3), (2, 4, 6), (1, 1, 1) and b = (6, 12, 3)ᵀ. A technician attempts to solve by computing the inverse of M and reports failure because 'the calculator gives an error.' A supervisor insists the system must have a solution because the physical network is running. You must diagnose why M has no inverse, determine whether the system is consistent, and describe the full solution set if it is.
1. Show that M is singular and explain what singularity means for the existence and uniqueness of solutions.
Read the worked answer
Compute det M by expansion along the first row:
det M = 1·(4·1 − 6·1) − 2·(2·1 − 6·1) + 3·(2·1 − 4·1) = 1·(4 − 6) − 2·(2 − 6) + 3·(2 − 4) = 1·(−2) − 2·(−4) + 3·(−2) = −2 + 8 − 6 = 0.
So det M = 0 and M is singular; it has no inverse. Note also that row 2 = 2 × row 1, so the rows are linearly dependent, confirming rank(M) < 3.
Singularity means the system Mx = b either has no solution or infinitely many; it cannot have a unique solution. The technician's calculator error is expected: inverting a singular matrix is impossible.
2. Determine whether the system is consistent. If it is, give the full solution set in parametric form.
Read the worked answer
Reduce the augmented matrix [M | b]:
Row 1: (1, 2, 3 | 6) Row 2: (2, 4, 6 | 12) Row 3: (1, 1, 1 | 3)
Row 2 − 2×Row 1 = (0, 0, 0 | 0), so row 2 is redundant and consistent with row 1 (12 = 2×6).
Row 3 − Row 1 = (0, −1, −2 | −3), i.e. y + 2z = 3.
From row 1: x + 2y + 3z = 6.
Let z = t (free parameter). Then y = 3 − 2t. Substitute: x + 2(3 − 2t) + 3t = 6, so x + 6 − 4t + 3t = 6, giving x = t.
Solution set: (x, y, z) = (t, 3 − 2t, t) for t ∈ ℝ, i.e. (0, 3, 0) + t(1, −2, 1).
The system is consistent with infinitely many solutions forming a line. The supervisor is right that a solution exists, but wrong to expect it to be unique. The physical interpretation: the flows are not pinned down by the equations alone; an extra constraint (e.g. fixing one flow) is needed for a unique answer.