Home

Solving Systems with Matrices

Linear-system matrix solver. Enter A and b to classify and solve Ax=b—unique, none or infinitely many—with steps.

展开更多 ▾
Ax = b
space-separated entries, semicolon-separated rows, e.g. 1 2;3 4
space-separated, e.g. 1 2

📖 Tutorial｜Solving Systems with Matrices

1. Definition
Write an m-equation n-unknown system as Ax = b and solve by Gaussian elimination on the augmented matrix.
2. Symbols
SymbolMeaning
Acoefficient matrix
xvector of unknowns
bRHS vector
3. How this tool works
  • Apply elementary row operations to [A|b];
  • Reduce to echelon form and back-substitute;
  • 0=nonzero means no solution; 0=0 means infinitely many.
4. Steps
  1. Enter A;
  2. Enter b (space-separated);
  3. Click Calculate for the solution and row steps.
5. Example
Example: 2x₁+x₂=1, x₁−x₂=2. Solution: x₁=1, x₂=−1.
6. Pitfalls
Columns of A equal the number of unknowns;
Length of b equals the number of equations;
No/infinite solutions are reported.

❓ FAQ｜Solving Systems with Matrices

What is the augmented matrix?
b appended as the last column of A, giving [A|b].
When is there no solution?
When echelon form has a 0=nonzero row.
When are there infinitely many?
When there are 0=0 rows and free variables.
Unique solution condition?
Square A with det A ≠ 0.
How to enter b?
Space-separated, e.g. 1 2 3.