The formula this page uses
Ax = 0 always has x = 0. Nontrivial solutions exist exactly when det A = 0, that is when rank(A) < number of unknowns.
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Trivial Solution Calculator
Solve trivial solution problems with clear steps, notation, and a final check.
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Linear algebra
Trivial Solution Calculator explained
The short version
- A homogeneous system is one where every equation equals zero: Ax = 0.
- Setting every variable to zero always works, and that is called the trivial solution.
- The real question is whether any other solution exists, and the determinant answers it.
What each part means
SymbolWhat it means
A — Coefficient matrixThe numbers in front of the unknowns. The right-hand side is all zeros.
x = 0 — Trivial solutionEvery unknown set to zero. It satisfies every homogeneous system, always.
rank(A) — RankThe number of independent equations left after row reduction.
n − rank(A) — Free variablesHow many parameters the solution set needs. Zero free variables means the trivial solution is the only one.
Show your work: a full example
- The homogeneous system2x + 3y = 0 and 4x + 6y = 0
- Determinant of the coefficient matrix(2)(6) − (3)(4) = 12 − 12 = 0
- Zero determinant, so nontrivial solutions existrow reduce to see them
- Row 2 minus 2 × Row 10x + 0y = 0, which tells you nothing new
- One real equation is left2x + 3y = 0, so x = −1.5y
- Write the whole solution set with a parameterlet y = 2t, then x = −3t, so every solution is t(−3, 2)
- Check with t = 12(−3) + 3(2) = 0 ✓ and 4(−3) + 6(2) = −12 + 12 = 0 ✓, and t = 0 gives back the trivial solution
A second, different case
- A different case: only the trivial solution2x + 3y = 0 and 4x − y = 0
- Determinant(2)(−1) − (3)(4) = −2 − 12 = −14
- Not zero, so no free variables existthe only solution should be x = y = 0
- Confirm by substitution, from the second equationy = 4x
- Put that into the first2x + 3(4x) = 14x = 0
- Solvex = 0, and therefore y = 0
- Picture itthese are two different lines through the origin, and two different lines cross at exactly one point
Copy-ready example
2x + 3y = 0 and 4x + 6y = 0
The homogeneous system
Questions about the Trivial Solution Calculator
Why is the zero solution called trivial?
Because it takes no work and it tells you nothing. Every homogeneous system has it, so finding it is never the point. The question a problem is really asking is whether anything else also works.
When is a nontrivial solution guaranteed without any calculation?
When there are more unknowns than equations. Three unknowns and one equation leaves two free variables no matter what the numbers are, which is the last row of the table above.
How does this connect to linear independence?
Directly. The columns of A are linearly independent exactly when Ax = 0 has only the trivial solution. A nontrivial solution is a recipe for combining the columns into zero, which is what dependence means.
What changes if the right-hand side is not zero?
Everything. A non-homogeneous system can be inconsistent and have no solutions at all, which a homogeneous one never can. Its full solution is one particular solution plus the homogeneous solution set found here.