What is Cramer's rule?

Cramer's rule is a formula for the solution of a square system of linear equations A x = b when the determinant of A is not zero. Each unknown x_i equals the determinant of the matrix obtained by replacing column i of A with b, divided by the determinant of A. It gives every unknown directly, without row reduction.

When does Cramer's rule not work?

It requires a square system with a non-zero coefficient determinant. If det A is zero, every formula divides by zero and the rule does not apply; the system then has either no solution or infinitely many, and which one depends on b, which row reduction of the augmented matrix decides. The rule also says nothing about systems with more or fewer equations than unknowns.

Why does replacing a column by b give x_i times det A?

The solution expresses b as a combination of the columns of A with coefficients x_1, …, x_n. A determinant is linear in each column, so replacing column i by that combination splits the determinant into n pieces, one per term. Every piece except the x_i piece has two equal columns and is zero, and the x_i piece is x_i times det A. Dividing by det A isolates x_i.

Is Cramer's rule a good way to solve systems?

For 2×2 and 3×3 systems by hand it is quick and easy to check. For larger systems it is far too slow: it needs n + 1 determinants of size n, each costing about n! operations by expansion, while Gaussian elimination solves the whole system in about two-thirds of n cubed operations. Its lasting value is as a formula and a proof of uniqueness, not as an algorithm.

What does Cramer's rule give for a homogeneous system?

If b is the zero vector, every replaced matrix has a column of zeros and every replaced determinant is zero, so every unknown is zero. That is the trivial solution, and when det A is not zero it is the only solution. Non-trivial solutions of A x = 0 exist exactly when det A is zero, which is the case the rule excludes.

Cramer's Rule Visualizer | Solve A x = b with Determinants

Key Terms

Cramer's rule — for a square system

A\mathbf{x} = \mathbf{b}

with

\det A \neq 0

,

x_i = \dfrac{\det A_i}{\det A}

, where

A_i

is

A

with its

i

-th column replaced by

\mathbf{b}

.

Coefficient matrix —

A

, holding the coefficients of the unknowns, one row per equation.

Right-hand side —

\mathbf{b}

, the constants; shown in amber throughout the tool.

Replaced matrix —

A_i

, formed by swapping

\mathbf{b}

into column

i

of

A

.

Unique solution — exactly one

\mathbf{x}

; for a square system this happens exactly when

\det A \neq 0

.

Singular system —

\det A = 0

; the system has no solution or infinitely many, and Cramer's rule does not apply.

Homogeneous system —

\mathbf{b} = \mathbf{0}

; every

\det A_i

is then

0

, and the unique solution is

\mathbf{x} = \mathbf{0}

.

Cost —

n + 1

determinants of size

n

, which grows far faster than the cost of elimination.

Getting Started with the Visualizer

Choose a system, then watch each unknown emerge as a ratio of two determinants.

• Use the Preset pills for five systems: a

2 \times 2

and a

3 \times 3

with integer solutions, one with fractional answers, a homogeneous one, and a singular one where the rule stops
• Use the Size stepper for

2

or

3

equations; resizing keeps the existing entries and pads with the identity
• Edit any entry of

A

or of

\mathbf{b}

directly, or press Shuffle for a random system built to have a small integer solution
• Hover the ? icon for a reminder of the formula and its precondition
• Press play or step manually; the step log on the right lists every determinant

The first scene after the system is always

\det A

, because everything depends on it. If it is zero the run ends there with an explanation; otherwise one scene per unknown follows, and the last scene checks the solution by multiplying it back.

How the Rule Runs

The visualizer follows the formula exactly.

• System —

A\mathbf{x} = \mathbf{b}

with the unknowns still symbolic
• Determinant —

\det A

is evaluated; a non-zero value licenses the rest
• Replace — for each

i

, the matrix

A_i

is shown with

\mathbf{b}

in column

i

, its determinant is evaluated, and

x_i = \det A_i / \det A

• Done — the solution vector fills, and

A\mathbf{x}

is recomputed to confirm it equals

\mathbf{b}

If

\det A = 0

the run stops after the determinant scene. The tool does not decide between no solution and infinitely many, because Cramer's rule cannot; that question belongs to row reduction of the augmented matrix.

The Opening Scene: The System

The player opens with

A\mathbf{x} = \mathbf{b}

written out: the coefficient matrix, the vector of unknowns still symbolic and greyed, and

\mathbf{b}

in amber. At the default preset the system is

3 \times 3

.

Nothing is computed yet. What the scene establishes is the shape: as many equations as unknowns, which is the only case the rule addresses.

Opening scene, frozen

A x = b with the unknowns still symbolic and b in amber. As many equations as unknowns - the only shape the rule addresses, because only square matrices have determinants.

The square shape matters because the rule divides by

\det A

, and only square matrices have determinants. A system with more equations than unknowns, or fewer, needs other methods.

The colour of

\mathbf{b}

is carried through the run: whenever it appears inside a replaced matrix, its column keeps the amber, so the swap is visible at a glance.

The Determinant of A

The first computation is

\det A

, evaluated by cofactor expansion and written into its slot. Everything else divides by this number.

The frozen picture below shows the default preset's

\det A = 7

, highlighted.

Determinant of A, frozen

det A = 7 evaluated and highlighted. Non-zero, so the columns are independent, the solution is unique, and every unknown is well defined.

A non-zero value here is the whole precondition of the rule and the whole content of the statement that the system has exactly one solution. It says the columns of

A

are independent, so

\mathbf{b}

can be written as a combination of them in exactly one way.

The tool evaluates the determinant numerically; the determinant visualizer shows the same expansion symbolically. The number that comes out is also the volume-scaling factor of

A

as a transformation, which is why a zero means collapse.

Replacing a Column

For each unknown

x_i

, the tool forms

A_i

by swapping

\mathbf{b}

into column

i

of

A

, evaluates

\det A_i

, and divides by

\det A

.

The frozen picture below is the second unknown of the default preset:

A_2

with

\mathbf{b}

in its middle column,

\det A_2 = 14

, and

x_2 = 14 / 7 = 2

.

Second unknown, frozen

A₂ with b swapped into its middle column, det A₂ = 14, and x₂ = 14 / 7 = 2. Linearity in the swapped column is why det A₂ equals x₂ det A.

The replaced determinant is

x_i \det A

, and the picture shows why. Column

i

of

A_i

is

\mathbf{b}

, which equals

x_1 \mathbf{a}_1 + \cdots + x_n \mathbf{a}_n

; a determinant is linear in each column, so

\det A_i

splits into

n

pieces, and every piece except the

x_i \mathbf{a}_i

one has two equal columns and vanishes.

Each unknown costs a fresh

n \times n

determinant. Nothing from

\det A_1

helps with

\det A_2

, which is the reason the method scales so badly and elimination does not.

When the Determinant Is Zero

On the singular preset the columns of

A

are proportional,

\det A = 0

, and the run stops after the determinant scene.

The frozen picture below is that stop: the system with its coefficient columns muted and no solution vector.

Singular system, frozen

The singular preset stopped: det A = 0, the coefficient columns muted, no solution vector. The rule cannot say whether there are no solutions or infinitely many; row reduction can.

The rule does not fail quietly here; it fails to exist, because every

x_i

would be a division by zero. Geometrically the two columns lie on one line, so the combinations

x_1 \mathbf{a}_1 + x_2 \mathbf{a}_2

only ever reach that line, and

\mathbf{b}

is either on it, with infinitely many ways to reach it, or off it, with none.

Which of the two holds is a question about

\mathbf{b}

, not about

A

, and determinants of

A

cannot answer it. Row reducing the augmented matrix

[A \mid \mathbf{b}]

can, and that is the tool to reach for when this one stops.

The Solution

The final scene fills the solution vector next to

A

and

\mathbf{b}

, and the caption multiplies

A\mathbf{x}

back out to confirm it equals

\mathbf{b}

.

The frozen picture below is the default preset solved:

\mathbf{x} = (1, 2, 3)

.

Solution, frozen

x = (1, 2, 3) in place next to A and b. Four determinants for three unknowns, none sharing work - a formula, not an algorithm.

The check is not decoration. Each unknown was computed independently from its own determinant, so an arithmetic slip in one of them shows up as a mismatch in one equation, and the check catches it.

Read across the whole run and the rule's character is clear:

n + 1

determinants, no sharing of work between them, and a formula at the end. It is the right tool for a

2 \times 2

solved on paper and the right theorem for understanding why the solution is unique; for anything larger, the elimination tools in this section do the same job with a fraction of the arithmetic.

Reading the Scene Player

Each scene shows the matrices involved in one step, with the arithmetic in the caption.

• In the determinant scene, all of

A

is highlighted and the

\det A

slot fills, accent if non-zero and muted if zero
• In a replace scene,

A_i

is drawn between determinant bars with the swapped column in amber and highlighted; the

\det A_i

slot is primary, the

\det A

slot secondary, and the

x_i

slot accent, with an arrow from the swapped column to its determinant
• In the done scene, the solution vector is highlighted next to

A

and

\mathbf{b}

• Fractions are shown as

a/b

in a smaller font; the unknowns not yet found are shown in grey as

x_1, x_2, \ldots

• The step log on the right keeps a record of every determinant evaluated

Choosing a System

The five presets each make a different point.

• $2 \times 2$ , integers — two equations, three determinants, solution

(1, 3)

; the whole rule in its smallest form
• $3 \times 3$ , integers — four

3 \times 3

determinants, solution

(1, 2, 3)

; the default, and the size at which the cost of the method starts to show
• $2 \times 2$ , fractions — the same rule with a non-integer answer,

(7/5, 19/10)

; the determinants are integers, the ratios are not
• Homogeneous —

\mathbf{b} = \mathbf{0}

, so every replaced matrix has a zero column and every

x_i = 0

• Singular — the columns of

A

are proportional,

\det A = 0

, and the run stops with the explanation

Shuffle builds its systems backwards: it picks a small integer solution first and computes

\mathbf{b}

from it, so the answers stay readable.

What Cramer's Rule Is

For a square system

A\mathbf{x} = \mathbf{b}

with

\det A \neq 0

, each unknown is a ratio of determinants:

x_i = \frac{\det A_i}{\det A}, \qquad A_i = A \text{ with column } i \text{ replaced by } \mathbf{b}

Why it works: the solution satisfies

\mathbf{b} = x_1 \mathbf{a}_1 + \cdots + x_n \mathbf{a}_n

, a combination of the columns of

A

. Put that combination into column

i

of

A

and expand the determinant by linearity in that column: every term except the

x_i \mathbf{a}_i

term has a repeated column and vanishes, and what remains is

x_i \det A

. So

\det A_i = x_i \det A

, and dividing gives the formula.

The same argument shows why

\det A \neq 0

is exactly the right condition. When it holds, the columns are independent and every

\mathbf{b}

is reached by exactly one combination; when it fails, they are dependent, and

\mathbf{b}

is reached either not at all or in infinitely many ways.

Cramer's rule is a theorem about the solution, not a recommended way to compute it. For

n

unknowns it needs

n + 1

determinants, and computing determinants by expansion costs

n!

operations each. Its value is that it is a closed formula: it shows that each

x_i

is a ratio of polynomials in the entries, which matters for symbolic work, for sensitivity analysis, and for small systems solved by hand. For solvability in general, see the linear systems theory page; for the rule among the other uses of the determinant, see the determinant applications page.

Key Properties

The rule inherits everything from the determinant.

• Precondition:

\det A \neq 0

; equivalently

A

invertible, equivalently the columns independent
• Uniqueness: when the rule applies, the solution it gives is the only one
• Homogeneous case:

\mathbf{b} = \mathbf{0}

forces every

\det A_i = 0

and hence

\mathbf{x} = \mathbf{0}

; non-trivial solutions of

A\mathbf{x} = \mathbf{0}

exist only when

\det A = 0

• Linearity in $\mathbf{b}$ : each

x_i

is a linear function of

\mathbf{b}

, which is the statement

\mathbf{x} = A^{-1}\mathbf{b}

written entry by entry
• Relation to the inverse: the rule is the adjugate formula

A^{-1} = \operatorname{adj} A / \det A

applied to

\mathbf{b}

• Scaling: multiplying an equation by a constant scales

\det A

and every

\det A_i

by the same factor, leaving the ratios unchanged
• Cost:

n + 1

determinants of size

n

; by expansion that is

(n+1)\,n!

operations against roughly

\tfrac{2}{3}n^3

for elimination

Why It Matters

Cramer's rule is used where a formula is worth more than speed.

• Small systems by hand: for

2 \times 2

and

3 \times 3

it is fast and self-checking, and it is the method behind many textbook shortcuts
• Symbolic solutions: when the entries are parameters rather than numbers, the rule gives each unknown as an explicit rational function of them
• Sensitivity: because each

x_i

is a ratio of determinants, the effect of perturbing one coefficient can be read off directly
• Theory: it is the shortest proof that a square system with non-zero determinant has exactly one solution, and it underlies the adjugate formula for the inverse
• Geometry: in the plane,

x_1

and

x_2

are ratios of parallelogram areas, which is where the rule came from historically
• The negative lesson: it is the standard example of a correct formula that is the wrong algorithm; solvers use elimination, and the rule explains why the answer exists

Worked Example

Take the default preset,

\begin{pmatrix} 1 & 1 & 1 \\ 2 & -1 & 1 \\ 1 & 2 & -1 \end{pmatrix} \mathbf{x} = \begin{pmatrix} 6 \\ 3 \\ 2 \end{pmatrix}

Determinant of $A$ , expanding along the first row:

1(1 - 2) - 1(-2 - 1) + 1(4 + 1) = -1 + 3 + 5 = 7

. Non-zero, so the rule applies.

First unknown: replace column

1

by

\mathbf{b}

.

\det A_1 = \begin{vmatrix} 6 & 1 & 1 \\ 3 & -1 & 1 \\ 2 & 2 & -1 \end{vmatrix} = 6(1 - 2) - 1(-3 - 2) + 1(6 + 2) = -6 + 5 + 8 = 7, \qquad x_1 = 7/7 = 1

Second unknown: replace column

2

.

\det A_2 = \begin{vmatrix} 1 & 6 & 1 \\ 2 & 3 & 1 \\ 1 & 2 & -1 \end{vmatrix} = 1(-3 - 2) - 6(-2 - 1) + 1(4 - 3) = -5 + 18 + 1 = 14, \qquad x_2 = 14/7 = 2

Third unknown: replace column

3

.

\det A_3 = \begin{vmatrix} 1 & 1 & 6 \\ 2 & -1 & 3 \\ 1 & 2 & 2 \end{vmatrix} = 1(-2 - 6) - 1(4 - 3) + 6(4 + 1) = -8 - 1 + 30 = 21, \qquad x_3 = 21/7 = 3

Check:

1 + 2 + 3 = 6

,

2 - 2 + 3 = 3

,

1 + 4 - 3 = 2

. Four

3 \times 3

determinants for three unknowns; elimination would have needed a handful of row operations.

Common Mistakes

A few mistakes recur.

• Replacing a row instead of a column —

\mathbf{b}

goes into column

i

; the columns of

A

are the coefficient vectors of the unknowns, and

x_i

multiplies column

i

• Forgetting to check $\det A$ first — if it is zero every

x_i

is a division by zero, and no amount of determinant arithmetic will rescue it
• Reading $\det A = 0$ as "no solution" — it means no unique solution; the system may still be consistent with infinitely many, which row reduction decides
• Dividing the wrong way —

x_i = \det A_i / \det A

, the replaced determinant over the original
• Using it on non-square systems — the rule needs as many equations as unknowns; other systems have no coefficient determinant to speak of
• Using it for large systems — correct but ruinously slow; it is a formula for understanding and for small cases, not a solver

Related Concepts

Determinant — the rule is made of determinants; the determinant tool shows how each one is evaluated.

Matrix inverse — the adjugate formula is Cramer's rule for every possible

\mathbf{b}

at once.

Gaussian elimination — the practical way to solve the same system, and the way to classify it when

\det A = 0

.

Rank — full rank is the same as

\det A \neq 0

for a square matrix.

Linear independence — of the columns of

A

, the geometric content of the precondition.

Homogeneous systems — the case

\mathbf{b} = \mathbf{0}

, where the rule gives only the trivial solution.

Solution sets — unique, none, or infinitely many, of which the rule handles the first.

Cramer's Rule

Step explanations

Key Terms

Getting Started with the Visualizer

How the Rule Runs

The Opening Scene: The System

The Determinant of A

Replacing a Column

When the Determinant Is Zero

The Solution

Reading the Scene Player

Choosing a System

What Cramer's Rule Is

Key Properties

Why It Matters

Worked Example

Common Mistakes

Related Concepts