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Glacius
Linear algebraConcept reference

Inconsistent systems

An inconsistent system has no solution; its row-reduced augmented matrix contains zero coefficients with a nonzero right-hand side.

1 min read1 figureSources & further reading
On this page 7 sections
  1. Overview
  2. A solution must satisfy every row
  3. Sometimes the right-hand side contains a fixed measurement
  4. Count contradictions, not equations
  5. Key takeaway
  6. Sources & further reading
  7. Concept connections

01A solution must satisfy every row#

A solution must satisfy every row. A row 0x+0y=30x+0y=3 demands 0=30=3, regardless of the variable values. One such contradiction rules out every candidate.

A row (0,0|3) means 0x+0y=3, or 0=3: impossible for every x and y. A row (0,0|0) means 0=0: always true and imposes no additional constraint.A row (0,0|3) means 0x+0y=3, or 0=3: impossible for every x and y. A row (0,0|0) means 0=0: always true and imposes no additional constraint.
Figure 1A row (0,0|3) means 0x+0y=3, or 0=3: impossible for every x and y. A row (0,0|0) means 0=0: always true and imposes no additional constraint.
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Read the last column as the right-hand side. In (0,0∣1)(0,0\mid1), both variable coefficients vanish but the 1 remains. The equation is impossible. In (0,0∣0)(0,0\mid0), it is always true.

Check your reasoning

Consistent? [102001]\left[\begin{array}{rr|r}1&0&2\\0&0&1\end{array}\right]

  1. ANo solution.
  2. BConsistent.
  3. CUse row 1 only.
Show answer and explanation
No solution.

The last row says 0=10=1.

02Sometimes the right-hand side contains a fixed measurement#

Sometimes the right-hand side contains a fixed measurement. If the residual row is 0x+0y=k−40x+0y=k-4, consistency requires k−4=0k-4=0, so the measurement must be 4.

Check your reasoning

A reading kk leaves 0x+0y=k−50x+0y=k-5. Other rows are consistent. Which kk avoids contradiction?

Show answer and explanation
5

The right side must be zero: k=5k=5.

03Count contradictions, not equations#

Count contradictions, not equations. Three rows can be consistent in two unknowns: x=2x=2, y=3y=3, and 0=00=0 all hold at (2,3)(2,3). A zero coefficient by itself is harmless; every coefficient must vanish before the row reads 0=b0=b.

Check your reasoning

Claim: three rows force no solution. Correct it: x=2x=2; y=3y=3; 0=00=0.

  1. ANo solution: 3 rows.
  2. B(2,3)(2,3) is a solution.
  3. CNo solution: 0=00=0.
Show answer and explanation
(2,3)(2,3) is a solution.

0=00=0 adds no restriction.

Key takeaway

Read the whole row: zero coefficients with a nonzero right-hand side mean no solution.

  • Detect a contradictory row in an augmented system.

Sources & further reading

  1. [1]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Inconsistent systems.” Math behind ML. /learn/la-consistency