In one line
Compare the rank of \(A\) with the rank of the augmented matrix \([A\,|\,B]\): unequal ranks mean no solution; equal ranks \(=n\) mean a unique solution; equal ranks \(<n\) mean infinitely many solutions.
Explained simply
Three equations in three unknowns may describe three planes. They can meet at one point (unique solution), along a line (infinitely many), or have no common point (no solution, inconsistent). Rank tells you which case you are in, without drawing anything.
Technically, for \(AX=B\) with \(n\) unknowns: if \(\rho(A)\ne\rho([A|B])\), the system is inconsistent. If \(\rho(A)=\rho([A|B])=r\), it is consistent; the solution is unique when \(r=n\) and has \(n-r\) free parameters when \(r<n\). This is the Rouché–Capelli criterion.
How it works
- Write the augmented matrix \([A|B]\).
- Reduce it to echelon form using row operations only.
- Read \(\rho(A)\) (ignore the last column) and \(\rho([A|B])\).
- Decide the case. If consistent, back-substitute (with parameters if \(r<n\)).
- With unknown constants \(\lambda,\mu\), find values making the last row \(0=0\) (infinite), \(0=\text{non-zero}\) (none), or keeping full rank (unique).
Key formula
| Condition | Result |
|---|---|
| \(\rho(A)\ne\rho([A|B])\) | No solution |
| \(\rho(A)=\rho([A|B])=n\) | Unique solution |
| \(\rho(A)=\rho([A|B])=r<n\) | Infinitely many, \(n-r\) parameters |
Quick example
Solve \(x+y+z=6\), \(x+2y+3z=14\), \(x+4y+7z=30\).
\(\left[\begin{array}{ccc|c}1&1&1&6\\1&2&3&14\\1&4&7&30\end{array}\right]\xrightarrow{R_2-R_1,\ R_3-R_1}\left[\begin{array}{ccc|c}1&1&1&6\\0&1&2&8\\0&3&6&24\end{array}\right]\xrightarrow{R_3-3R_2}\left[\begin{array}{ccc|c}1&1&1&6\\0&1&2&8\\0&0&0&0\end{array}\right]\).
\(\rho(A)=\rho([A|B])=2<3\): infinitely many solutions. Let \(z=t\): \(y=8-2t\), \(x=6-y-z=t-2\).
Check third equation: \((t-2)+4(8-2t)+7t=30\). If the last constant were 31 instead of 30, the last row would read \(0=1\): inconsistent.
Common mistakes
- Doing column operations on the augmented matrix; only row operations are allowed here.
- Finding \(\rho(A)\) and forgetting \(\rho([A|B])\).
- Stating "infinitely many solutions" without giving the general solution.
How MU asks it
Very common 6-mark question: "Test for consistency and solve", or "For what values of \(\lambda\) and \(\mu\) does the system \(x+y+z=6\), \(x+2y+3z=10\), \(x+2y+\lambda z=\mu\) have (i) no solution, (ii) a unique solution, (iii) infinitely many solutions?" (answers: \(\lambda=3,\mu\ne10\); \(\lambda\ne3\); \(\lambda=3,\mu=10\)). Show the augmented matrix, operations, both ranks, the conclusion and the solution.