Consistency of a system and the number of solutions
A system is consistent if it has at least one solution and inconsistent if it has none. Test |A| first, then (adj A)B.
What consistent means
A system of linear equations is consistent if it has at least one solution, and inconsistent if it has none. Think of two equations in x and y as two lines. If they cross once, there is a unique solution. If they are the same line, there are infinitely many. If they are parallel, there is no solution. With three variables, the matrix test below decides.
Decision table for AX = B
| |A| | (adj A)B | Number of solutions | Type |
|---|---|---|---|
| ≠ 0 | Not needed | Exactly one, X = A⁻¹B | Consistent |
| = 0 | ≠ O | None | Inconsistent |
| = 0 | = O | Infinitely many, or none | Consistent if infinitely many; inconsistent if none |
Example 1: a system with no solution
Check whether the system 2x − y = 5, 4x − 2y = 7 is consistent.
- A = [2, −1; 4, −2], X = [x; y], B = [5; 7]
- |A| = 2 × (−2) − (−1) × 4 = −4 + 4 = 0, so there is no unique solution.
- adj A = [−2, 1; −4, 2]
- (adj A)B = [−2 × 5 + 1 × 7; −4 × 5 + 2 × 7] = [−3; −6]
- (adj A)B ≠ O, so the system has no solution.
- On a graph the lines are parallel: 2x − y = 5 and 2x − y = 3.5.
The system is inconsistent (no solution).
Example 2: a system with infinitely many solutions
Examine the system x + y + z = 3, x − y + z = 1, 2x + 2z = 4.
- A = [1, 1, 1; 1, −1, 1; 2, 0, 2], B = [3; 1; 4]
- |A| = 1(−2 − 0) − 1(2 − 2) + 1(0 + 2) = −2 − 0 + 2 = 0
- adj A = [−2, −2, 2; 0, 0, 0; 2, 2, −2]
- (adj A)B = [−6 − 2 + 8; 0; 6 + 2 − 8] = [0; 0; 0] = O
- So compare the equations: the third is the sum of the first two, so it adds nothing new.
- Subtracting the second from the first: 2y = 2, so y = 1 and x + z = 2. Put x = k: z = 2 − k.
Consistent, with infinitely many solutions: x = k, y = 1, z = 2 − k for any real k.
Watch out
- Find |A| first. If |A| ≠ 0, stop: the system is consistent with one solution.
- If |A| = 0, you must work out (adj A)B before deciding.
- (adj A)B = O does not prove infinitely many solutions. Solve or compare the equations to be sure.
- Inconsistent means no solution; it never means 'many solutions'.
- Write B as a column matrix, so that (adj A)B is defined.
If two bills for the same items give equations with no common solution, the system is inconsistent, so at least one bill is wrong. Accountants and shopkeepers catch billing errors with exactly this kind of check.
2 marks: “Examine the consistency of the system 2x + 3y = 5, 6x + 9y = 12.” |A| = 18 − 18 = 0. adj A = [9, −3; −6, 2]. (adj A)B = [45 − 36; −30 + 24] = [9; −6] ≠ O. So the system is inconsistent.
2 marks: “Examine the consistency of the system x + y = 5, 2x + 2y = 10.” |A| = 2 − 2 = 0. adj A = [2, −1; −2, 1]. (adj A)B = [10 − 10; −10 + 10] = O. The second equation is twice the first, so there are infinitely many solutions: the system is consistent.
1 marks: “For what values of k does the system kx + 2y = 5, 3x + y = 1 have a unique solution?” A unique solution needs |A| ≠ 0: k − 6 ≠ 0. So every real k except k = 6.
|A| ≠ 0: one answer. |A| = 0: work out (adj A)B; if it is not zero, there is no answer.