Cramer's Rule Calculator

Cramer's rule solves a system by determinants alone: each unknown is its own determinant divided by the determinant of the system.

Solution

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Solution—

Δ of the system—

Δx—

Δy—

Δz—

Type of system—

How it was solved

The formula

xᵢ = Δᵢ ÷ Δ

Why it works

Gabriel Cramer published the rule in 1750. Its appeal is that it gives each unknown independently, with a single formula and no elimination — you can solve for z without ever touching x or y, which no other method lets you do.

How to solve it by hand

Work out Δ, the determinant of the coefficient matrix
For each unknown, replace its column with the constants
Work out that determinant: Δx, Δy, Δz
Divide each by Δ

What is worth knowing

The rule's independence is genuinely useful when you need one unknown out of many, but as a general method it is a trap: solving an n×n system by Cramer's rule takes on the order of n! operations against n³ for elimination. For a 20×20 system that is the difference between a fraction of a second and longer than the age of the universe. It is also numerically fragile when the determinant is close to zero. Learn it for the insight into determinants, and reach for elimination when you actually have to solve something.

Frequently asked questions

When is Cramer's rule useful?

For small systems, and when you need only one unknown out of several. It gives each one independently.

Why is it not used for large systems?

The cost grows like n factorial against n cubed for elimination. Past a handful of unknowns it becomes hopeless.

What if the determinant is zero?

The rule does not apply. The system either has no solution or infinitely many, and you need another method to tell which.

Does it work for non-square systems?

No. It requires as many equations as unknowns, which is what makes the coefficient matrix square.