Analyze augmented matrices using structured row reduction. See ranks, pivots, solutions, and consistency decisions clearly. Build confidence through examples, exports, graphs, and plain guidance.
| Case | System | Rank(A) | Rank([A|b]) | Expected Result |
|---|---|---|---|---|
| Unique | x + y + z = 6, 2x − y + z = 3, x + 2y − z = 3 | 3 | 3 | Consistent with one exact solution |
| Infinite | x + y + z = 6, 2x + 2y + 2z = 12, x − y = 0 | 2 | 2 | Consistent with infinitely many solutions |
| Inconsistent | x + y + z = 6, 2x + 2y + 2z = 13, x − y = 0 | 2 | 3 | Inconsistent with no exact solution |
Let A be the coefficient matrix and [A|b] the augmented matrix. A system is consistent when rank(A) = rank([A|b]). It is inconsistent when rank(A) ≠ rank([A|b]).
If rank(A) = rank([A|b]) = n, the system has one unique solution. If rank(A) = rank([A|b]) < n, the system has infinitely many solutions.
Elementary row operations preserve the solution set. The calculator uses row swaps, row scaling, and row replacement to reach reduced row echelon form.
For square systems, a nonzero determinant confirms a unique solution. A zero determinant means the matrix may still be consistent, but rank testing is required.
A matrix system is useful only when its equations agree with each other. Consistency tells you whether the augmented matrix preserves the same structural information as the coefficient matrix. This calculator checks that relationship carefully. It shows whether a system has one solution, many solutions, or no exact solution.
The main test compares rank(A) with rank([A|b]). These ranks are found through row reduction. The calculator also computes the determinant for the square coefficient matrix. That extra value helps interpret singular and nonsingular systems. Every step is displayed so the logic remains transparent.
Row swaps, row scaling, and row replacement simplify the matrix without changing the solution set. Reduced row echelon form exposes pivots and free variables. If every variable column has a pivot, the solution is unique. If at least one variable is free and the ranks still match, the system stays consistent but produces infinitely many valid answers.
An inconsistent system creates a contradictory row after elimination. That row often looks like 0x + 0y + 0z = c, where c is not zero. This means the original equations cannot all be true at the same time. The augmented rank becomes larger than the coefficient rank, and the calculator marks the system as inconsistent.
Use it for algebra practice, linear systems homework, engineering models, and data setup checks. The example table helps you compare common cases quickly. The export options help save results for reports or revision notes. The graph gives a fast visual comparison between the variable count and both ranks.
A system is consistent when the coefficient rank equals the augmented rank. Then at least one exact solution exists. The solution may be unique or infinite.
An inconsistent system appears when rank([A|b]) becomes larger than rank(A). This means the equations contradict each other after row reduction.
The determinant is a quick square-matrix check. A nonzero value guarantees one unique solution. A zero value means you still need the rank test.
Yes. A zero determinant only means the coefficient matrix is singular. The system can still be consistent with infinitely many solutions if both ranks match.
The steps show how the original augmented matrix becomes simpler. They reveal pivots, free variables, and contradictions. This makes the final decision easier to trust.
This single-file version focuses on 2 × 2, 3 × 3, and 4 × 4 square systems. That keeps the interface simple and the explanations clear.
The calculator labels the system as consistent with infinitely many solutions. It also shows one valid sample solution produced by setting free variables to zero.
The graph compares the number of variables with rank(A) and rank([A|b]). That quick visual helps explain why the system is unique, infinite, or inconsistent.