System Of Equations Matrix Calculator
A system of three equations in three unknowns is the classic point where hand algebra starts to wobble: one sign slip in the elimination and the whole answer is wrong. This calculator solves 2×2 and 3×3 linear systems with Cramer’s rule, prints the determinant it divided by, then plugs the answers back into every equation so you can see each one balance. The worked 3×3 example below is pre-filled; press Solve system and the answers appear.
The solution is computed with Cramer’s rule and printed to four decimals. The check lines plug the answers back into each equation; every residual should read 0.000000, which is the tool’s way of proving its own answer.
Quick Answer
The pre-filled system x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2 solves to x = 1.0000, y = 2.0000, z = 3.0000. The coefficient matrix has determinant 7, so exactly one solution exists. Press Solve system in the calculator above and the check lines plug those answers back into each equation, with every residual reading 0.000000.
Key takeaways
- Cramer's rule solves a system as quotients of determinants: x = Dx ÷ D, y = Dy ÷ D, z = Dz ÷ D. The calculator runs that method for both 2×2 and 3×3 systems.
- A nonzero determinant means exactly one solution. A zero determinant means the tool cannot divide, so it refuses the answer instead of guessing.
- This page is the 3×3 matrix page. For a dedicated two-equation tool, see the linear systems calculator, which covers the 2×2 case on its own.
- New in this version: the 3-equation mode used to show a "coming soon" message. It is now a complete, working solver (see the version note below).
- Quick Answer
- How to use the system of equations matrix calculator
- The Cramer's rule formulas this calculator runs
- Worked example: the 3×3 system
- The 2-equation mode: a second worked example
- When the determinant is zero
- What competing calculators miss
- New in this version: the 3-equation mode finally works
- Matrix method vs elimination vs substitution
- Frequently asked questions
How to use the system of equations matrix calculator
Pick the system size, type the coefficients, and read the answer. The results stay hidden until you press Solve system, so nothing computes before you are ready.
- Choose 2 or 3 equations: the input rows switch automatically. In 2-equation mode the third equation and the z-column disappear.
- Enter the coefficients row by row: a₁, b₁, c₁ are the numbers in front of x, y and z in the first equation, and d₁ is the constant on the right. The 3×3 example (1, 1, 1, 6 / 2, −1, 1, 3 / 1, 2, −1, 2) is pre-filled.
- Press Solve system: the calculator prints x, y (and z), the determinant of the coefficient matrix, and one check line per equation.
- Read the check lines: each one substitutes the answers back into its equation. A correct solution leaves every residual at 0.000000.
- Copy or share with the buttons in the results box when you want to keep the worked solution.
Invalid entries are rejected outright: any blank or non-numeric field shows "Please fill all fields with valid numbers", and a zero determinant shows "No unique solution exists (determinant is zero)". Decimals and negatives are accepted everywhere. For a single equation in one unknown, the lighter three-step linear equations calculator is the faster tool.
The Cramer's rule formulas this calculator runs
Cramer's rule turns solving into computing determinants. D is the determinant of the coefficient matrix; Dx, Dy, Dz are the same matrix with one column replaced by the constants. Every solution is a quotient of two determinants.
x = Dx ÷ D = (c₁b₂ − c₂b₁) ÷ D y = Dy ÷ D = (a₁c₂ − a₂c₁) ÷ D
3×3 system: D = a₁(b₂c₃ − b₃c₂) − a₂(b₁c₃ − b₃c₁) + a₃(b₁c₂ − b₂c₁)
x = Dx ÷ D y = Dy ÷ D z = Dz ÷ D, each numerator determinant formed by swapping the constants into the matching column.
If you only need D itself, a dedicated 2×2 determinant calculator computes just that number. The matrix route has a cousin worth knowing: the same system can be written AX = B and solved as X = A⁻¹B, which needs the inverse first, something a 2×2 inverse matrix calculator supplies. Cramer's rule skips the inverse entirely, which is why this tool uses it.
Worked example: the 3×3 system
The calculator opens with this system pre-filled, and these are the exact numbers it produces. The four determinants below are the whole computation: three column-swaps and one division each.
| Determinant | Built from | Value | Gives |
|---|---|---|---|
| D | Coefficient matrix | 7 | Nonzero, so one solution exists |
| Dx | Constants in the x column | 7 | x = 7 ÷ 7 = 1.0000 |
| Dy | Constants in the y column | 14 | y = 14 ÷ 7 = 2.0000 |
| Dz | Constants in the z column | 21 | z = 21 ÷ 7 = 3.0000 |
The check lines then prove the answer: 1(1.0000) + 1(2.0000) + 1(3.0000) = 6.0000, 2(1.0000) − 1(2.0000) + 1(3.0000) = 3.0000, 1(1.0000) + 2(2.0000) − 1(3.0000) = 2.0000, every residual 0.000000.
The 2-equation mode: a second worked example
Switch the selector to 2 equations and type this system: 2x + y = 5 and x − y = 1. The determinant is D = 2(−1) − 1(1) = −3, and the tool returns x = 2.0000, y = 1.0000, with both check lines balancing at 5.0000 and 1.0000. This mode is identical to the standalone linear systems calculator: it survives there as its own 2×2 page, while this page's home turf is the 3×3 matrix case.
When the determinant is zero
A zero determinant means the equations do not pin down a single point, so the tool stops and says so instead of dividing by zero. Geometrically, at least two of the lines (or planes) are parallel or identical, and the system either has no solution at all or infinitely many. The tool does not try to tell those two cases apart or parametrize the infinite family; it reports "No unique solution exists (determinant is zero)" and leaves the classification to you. That refusal is deliberate: a wrong single answer is worse than an honest non-answer.
One precision note: the zero check fires only on an exactly zero determinant. A determinant that is merely tiny, like 0.0000001, is technically nonzero and will produce a valid but enormous answer, which is the mathematically correct behavior for a nearly parallel system. If your coefficients came from measurements, double-check the inputs before trusting a huge result.
What competing calculators miss
The 2×2/3×3 solver space is crowded, and each camp leaves something on the table:
- Step-by-step elimination tools (8gwifi's solver): show every row operation up to 10×10, which is excellent for learning. The price is complexity: a heavy interface for what is often a 30-second homework check.
- Single-method 3×3 pages (FIRGELLI's 3×3 calculator): print the Cramer's formulas and the determinant, matching our math exactly. But they are 3×3-only with no 2-equation mode and no answer verification.
- RREF classifiers (Softonic's solver): classify unique, none, and infinitely many solutions with residuals, which is genuinely stronger on the zero-determinant case. They stop at numeric values only, with no determinant readout and no plain-English reading of what happened.
- Method-choice solvers (MiniWebtool): offer elimination, determinant, or matrix steps for 2×2 through 4×4. Breadth over polish: the steps are terse and there is no back-substitution check.
- Exact-fraction solvers (OnlyCalculators): give fractions like 3/2 with plain-English steps, the friendliest teaching tool of the bunch. No determinant is shown, and decimals-only input can surprise students expecting fractions.
- The gap this page fills: a finished 3×3 Cramer's solver with the determinant always on display, a per-equation back-substitution check that proves the answer, and plain refusal instead of a guessed answer when D = 0.
New in this version: the 3-equation mode finally works
An honest changelog note. The previous version of this tool offered a "3 Equations (3 Variables)" option that only showed a "coming soon" message; worse, the first two equation rows had no constant inputs for the 3×3 case, so the mode could not have worked at all. This rewrite completes it: the third equation row is live, each of the first two rows gains its d constant, and the 3×3 system solves by Cramer's rule exactly like the 2×2 mode. The old article prose also claimed Gaussian elimination, matrix inverses and row-reduced output; none of that existed in the tool, and none of it is claimed here. What the tool does is Cramer's rule, a determinant readout, and a back-substitution check, and every word on this page describes exactly that.
Matrix method vs elimination vs substitution
Three roads reach the same answer, and they suit different moments. This tool takes the first road; knowing the other two helps you pick the right one next time.
Cramer's rule (this tool)
Best when you want the answer fast with the determinant as a sanity check. Determinants are the weak point: each one is a fresh chance at an arithmetic slip, which is why the tool shows all four and then verifies by substitution.
Elimination
Best for learning and for larger systems. Adding multiples of equations to kill variables is the most mechanical method by hand, and it is the one that scales to 4×4 and beyond where Cramer's rule gets unwieldy.
Substitution
Best for 2×2 systems where one equation already isolates a variable, like x = y + 1. For 3×3 it turns into nested fractions quickly and is the easiest method to botch.
Frequently asked questions
With Cramer's rule: it computes the determinant D of the coefficient matrix, then three more determinants with the constants column swapped into the x, y and z columns, and divides. For the pre-filled 3×3 example that is x = 7 ÷ 7, y = 14 ÷ 7, z = 21 ÷ 7, giving x = 1.0000, y = 2.0000, z = 3.0000.
Whether a unique solution exists. A nonzero determinant means the equations pin down exactly one point, so the tool divides and answers. A zero determinant means they do not, so the tool refuses the answer instead of guessing. The determinant is always printed under the solution.
Because dividing by zero is not a computation, and any single answer it printed would be a guess. A zero determinant means the system has no solution (parallel or contradictory equations) or infinitely many (dependent equations). The tool reports "No unique solution exists (determinant is zero)" and leaves the classification to you.
Back-substitution proof. The tool plugs its answers for x, y and z into each original equation and shows the residual, the difference between the two sides. Every residual should read 0.000000. If one does not, something went wrong with the inputs, and the check line tells you which equation to re-examine.
Yes. Every field accepts decimals, negatives and large values. The answers print to four decimals, so a result like −1.9091 is exact to that precision, and the check lines verify it the same way they verify whole numbers.
It was an unfinished stub: the option existed in the menu but the solver behind it was never written, and the first two rows lacked the constant inputs a 3×3 system needs. This version completes it. The 3-equation mode now runs the full Cramer's rule computation, exactly like the 2-equation mode.
Use this page for three equations in three unknowns, or when you want the determinant and the back-substitution check shown with the answer. Use the linear systems calculator for a plain two-equation solve; it is the dedicated 2×2 page and this page links to it for that job.
Yes. It runs entirely in your browser, needs no account, and stores nothing. Copy the result with “Copy my results” or send it to another app with “Share results”.
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References
- Eric W. Weisstein. Cramer's Rule. Wolfram MathWorld (definition of Cramer's rule: each variable equals a quotient of determinants, the denominator being the determinant of the coefficient matrix). Accessed October 24, 2026.
- Ken Kuttler et al. Application of the Determinant to Inverses; Cramer's Rule. LibreTexts, Linear Algebra: A First Course, De Anza College (formal statement x_i = det(A_i)/det(A) with the i-th column replaced by the constants). Accessed October 24, 2026.
- Paul Dawkins. Algebra: Systems of Equations. Paul's Online Math Notes, Lamar University (systems of two and three equations, substitution and elimination, inconsistent and dependent systems). Accessed October 24, 2026.
- Denny Burzynski and Wade Ellis. Solve Systems of Equations Using Determinants. LibreTexts, Beginning and Intermediate Algebra, Coastline College (step-by-step Cramer's rule procedure including the dependent/inconsistent classification when D = 0). Accessed October 24, 2026.
- Khan Academy. Systems of equations. Khan Academy Algebra unit (elimination and substitution methods for systems of equations). Accessed October 24, 2026.