Skip to content
Allin
Sign up · Sign in

Cramer's Rule Calculator

Solves a 2×2 or 3×3 system with exact fractions, shows every determinant expanded term by term, and when D = 0 separates the two cases most calculators merge: dependent with infinitely many solutions, or inconsistent with none.

Related tools

All Matrices & vectors tools →
Descartes' Rule of Signs CalculatorSign changes in f(x) and f(−x) marked one by one, then every possible (positive, negative, complex) combination that adds up to the degree.Simpson's Rule CalculatorIntegrate with the 1/3 and 3/8 rules, with the weight pattern shown, the subinterval count adjusted when it must be, and the error bound.Trapezoidal Rule CalculatorApproximates an integral from a function or from raw measured points, and unevenly spaced points are supported — real data rarely arrives on a regular grid. Draws the trapezoids, gives the error bound and compares to the exact value when one is obtainable.Rule of threeSolve a proportion: if A gives B, what does C give?Angle Between Two Vectors Calculatorθ = arccos(a·b / (|a||b|)) in 2D or 3D, in degrees and radians, with the dot product, both magnitudes, the projections and a parallel / perpendicular / acute / obtuse verdict.Column Space Calculator (basis of Col(A), rank, pivot columns)A basis of the column space of any matrix up to 6×6, taken from the ORIGINAL pivot columns rather than the reduced ones — with the rank, the free columns, the full RREF and every row operation.Cross Product Calculatora × b in 3D with the determinant expansion shown, the magnitude |a||b|sin θ, the parallelogram and triangle areas, the unit normal and the right-hand rule.Dot Product Calculatora·b in 2 to 10 dimensions with the term-by-term sum, both magnitudes as exact radicals, the angle in degrees and radians, the projections, and a figure. Parallel and orthogonal are decided in exact fractions, so (1,1,1)·(2,2,2) returns 0°, not NaN.

Cramer's Rule Calculator is free to use as often as you like, directly from this page. It sits under Matrices & vectors in our catalogue, alongside Descartes' Rule of Signs Calculator and Simpson's Rule Calculator.

How to use it

  1. Open the tool — no signup or install needed.
  2. Enter your input or adjust the available options.
  3. Get your result instantly, then copy or download it.

Frequently asked questions

What does Cramer's Rule Calculator do?

Solves a 2×2 or 3×3 system with exact fractions, shows every determinant expanded term by term, and when D = 0 separates the two cases most calculators merge: dependent with infinitely many solutions, or inconsistent with none.

When would I actually use this?

Checking a decomposition done by hand, solving a linear system, and finding the eigenvalues that describe how a transformation stretches space.

What is the most common mistake?

Inverting a matrix to solve a system. Elimination or a factorisation is both faster and far more stable numerically; an explicit inverse amplifies rounding error, especially when the matrix is close to singular.

How is Cramer's Rule Calculator different from Descartes' Rule of Signs Calculator?

They sit next to each other but answer different questions: Descartes' Rule of Signs Calculator is the one to open when you need it to sign changes in f(x) and f(−x) marked one by one, then every possible (positive, negative, complex) combination that adds up to the degree. Pick whichever matches what you're starting from — both are free.

Is there a tool for the next step?

Simpson's Rule Calculator is the closest one after this: Integrate with the 1/3 and 3/8 rules, with the weight pattern shown, the subinterval count adjusted when it must be, and the error bound.

What else is worth having open alongside it?

Trapezoidal Rule Calculator and Rule of three — they come up in the same task often enough to be worth a second tab.

Where do the figures come from?

The decompositions follow their textbook definitions with partial pivoting where it applies. Results are exact for small integer matrices and subject to floating-point rounding otherwise.

Further reading

All guides →
ExplainerThe Flat Pattern of a Cone Is a Sector, Not a CircleRoll a 50 mm cone 80 mm tall and its pattern is a 190.8° slice of a 94 mm circle. Flatten a nearly cylindrical lampshade and the pattern radius runs to 839 mm — the shallower the taper, the further away the apex.ExplainerA 1 % Drop Rate Does Not Mean One Hundred RunsAt 1 %, a hundred attempts give you 63.4 % — not certainty. Ninety per cent takes 230 attempts and ninety-nine takes 459, and more than a third of players are still empty-handed at a hundred.ExplainerFibonacci and the Golden RatioThe Fibonacci sequence adds each pair of terms to make the next; the ratio of neighboring terms homes in on the golden ratio φ ≈ 1.618. See how, and where the pattern turns up.ExplainerWhat Is a Z-Score? Standard Deviations Above the MeanA z-score is z = (x − μ) / σ — how many standard deviations a value sits from the mean. Learn how to compute it, read the standard normal curve, and turn it into a percentile.How-toHow to Find the Distance Between Two PointsUse the distance formula d = √((x₂−x₁)² + (y₂−y₁)²) to measure the straight-line distance between two points on a plane, with a worked example and the midpoint too.How-toHow to Calculate the Slope of a Line: Rise Over RunFind the slope of a line from two points with m = (y₂ − y₁) / (x₂ − x₁), and read what a positive, negative, zero, or undefined slope tells you.