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Nonhomogeneous Linear Systems

Nonhomogeneous-system classifier. Enter A and b to classify unique/infinite/no solution with rank conditions—for solution structure.

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Ax = b, rank(A) = rank([A|b]) consistent
space-separated entries, semicolon-separated rows, e.g. 1 2;3 4
space-separated

📖 Tutorial｜Nonhomogeneous Linear Systems

1. Definition
A system with b≠0, Ax=b, is nonhomogeneous. Its solvability depends on rank(A) vs rank([A|b]).
2. Symbols
SymbolMeaning
rank(A)rank of A
rank([A|b])rank of augmented matrix
nnumber of unknowns
3. How this tool works
  • rank(A) < rank([A|b]) ⇒ no solution;
  • rank(A)=rank([A|b])=n ⇒ unique solution;
  • rank(A)=rank([A|b])
4. Steps
  1. Enter A and b;
  2. Click Calculate to compare ranks and classify.
5. Example
Example: A=1 1;2 2, b=2 5. rank(A)=1, rank([A|b])=2, so no solution.
6. Pitfalls
No solution means no particular solution;
Infinite solutions = particular + homogeneous general solution;
Comparing ranks is the key to consistency.

❓ FAQ｜Nonhomogeneous Linear Systems

When is there no solution?
rank(A) < rank([A|b]).
When is the solution unique?
rank(A)=rank([A|b])=n.
When are there infinitely many?
rank(A)=rank([A|b])
Structure of a nonhomogeneous solution?
One particular solution + general solution of the homogeneous system.
How to enter b?
Space-separated.