In one line
\(AX=0\) always has the trivial solution \(X=0\); it has non-trivial (infinitely many) solutions exactly when \(\rho(A)<n\), the number of unknowns, i.e. \(|A|=0\) for a square system.
Explained simply
A homogeneous system has all right-hand sides zero. Setting every unknown to zero obviously works, so the system is always consistent. The interesting question is whether other solutions exist. If some equations repeat information (rank less than the number of unknowns), you have fewer real conditions than unknowns, so some unknowns can be chosen freely.
Technically, for \(AX=0\) with \(n\) unknowns and \(\rho(A)=r\): if \(r=n\), only \(X=0\); if \(r<n\), there are infinitely many solutions involving \(n-r\) free parameters.
How it works
- Write the coefficient matrix \(A\).
- Reduce \(A\) to echelon form by row operations and find \(r=\rho(A)\).
- If \(r=n\): only the trivial solution.
- If \(r<n\): choose \(n-r\) unknowns as parameters (\(t\), \(s\)), write the reduced equations, and solve upwards (back-substitution).
- If the system has a parameter \(\lambda\), set \(|A|=0\) to find the values of \(\lambda\) for non-trivial solutions.
Key formula
\(\rho(A)=n\Rightarrow X=0\) only; \(\ \rho(A)=r<n\Rightarrow\) infinitely many solutions with \(n-r\) free parameters. For square \(A\): non-trivial solution \(\iff|A|=0\).
Quick example
Solve \(x+2y+3z=0\), \(2x+4y+6z=0\), \(x+y+z=0\).
The echelon form of \(A\) is \(\begin{bmatrix}1&2&3\\0&-1&-2\\0&0&0\end{bmatrix}\), so \(r=2<3\): one free parameter.
Let \(z=t\). From \(-y-2z=0\): \(y=-2t\). From \(x+2y+3z=0\): \(x=4t-3t=t\).
Solution: \((x,y,z)=t(1,-2,1)\). Check in the third equation: \(t-2t+t=0\).
Common mistakes
- Saying a homogeneous system can be inconsistent; it never is.
- Giving one particular non-zero solution instead of the general solution with a parameter.
- Using the wrong number of free parameters; it is \(n-r\).
How MU asks it
5/6-mark parts: "Solve \(x+y-3z+2w=0\), ... " or "Find the values of \(\lambda\) for which the system \((3-\lambda)x+\dots=0\) has a non-trivial solution". State the rank condition, show the echelon form, the rank, the free parameters and the general solution, and verify in one equation.