Reduce matrices and reveal exact solution spaces. Track rank nullity pivots and free-variable relationships clearly. Generate exports and summaries for classes, homework, and reviews.
This sample system shows a 3 × 4 homogeneous matrix with one free variable, so the solution set forms a one-dimensional subspace.
| Equation | x1 | x2 | x3 | x4 | Right side |
|---|---|---|---|---|---|
| 1 | 1 | 2 | -1 | 0 | 0 |
| 2 | 2 | 4 | -2 | 1 | 0 |
| 3 | -1 | -2 | 1 | 3 | 0 |
Core matrix equation: A x = 0
Rank-nullity theorem: nullity(A) = number of variables − rank(A)
Decision rule: if rank(A) equals the number of variables, only the trivial solution exists. Otherwise, the system has infinitely many solutions, and the basis vectors describe the full solution space.
Method: the calculator converts the coefficient matrix to reduced row echelon form using Gaussian elimination, then reads pivot columns, free variables, parametric equations, and basis vectors directly from that final matrix.
It is a linear system where every equation equals zero. Such systems are always consistent because the zero vector always satisfies the equations.
The trivial solution, where every variable equals zero, always works. That makes every homogeneous system consistent, even when no nontrivial solution exists.
Rank counts the pivot columns in the coefficient matrix. It shows how many independent equations constrain the variables in the system.
Nullity is the number of free variables. It equals the number of variables minus the matrix rank and gives the dimension of the solution space.
Nontrivial solutions appear when the rank is smaller than the number of variables. Then at least one free variable exists, creating infinitely many solutions.
They are independent vectors that span the full solution space. Any solution can be written as a linear combination of those basis vectors.
Yes. The calculator accepts simple fractions such as 1/2, 3/4, or -5/6, along with ordinary decimal numbers.
Determinants are defined only for square matrices. For rectangular systems, rank and nullity provide the relevant structure needed for solving.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.