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Cramer's Rule Calculator

Solve systems of linear equations using Cramer's Rule. See each determinant matrix step by step -- det(A), det(A_x), det(A_y) -- with highlighted column substitutions.

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Simple 2x2 System

Solve 2x + 3y = 7 and x - y = 1 using Gaussian elimination

Key values: 2×2 system · Unique solution · x=2, y=1

3x3 Engineering System

Solve a 3-variable system commonly found in circuit analysis

Key values: 3×3 system · Cramer's Rule · 3 unknowns

Parallel Lines (No Solution)

Explore what happens when two equations have no intersection

Key values: 2×2 system · Inconsistent · No solution

Documentation

Cramer's Rule

Cramer's rule expresses each unknown as a ratio of determinants. For the system $A\mathbf{x} = \mathbf{b}$ :

x_i = \frac{\det(A_i)}{\det(A)}

where $A_i$ is the matrix $A$ with its ith column replaced by the constant vector $\mathbf{b}$ .

The 2×2 Case

For the system $a_1x + b_1y = c_1$ , $a_2x + b_2y = c_2$ :

x = \frac{\begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}} = \frac{c_1 b_2 - c_2 b_1}{a_1 b_2 - a_2 b_1}

y = \frac{\begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}} = \frac{a_1 c_2 - a_2 c_1}{a_1 b_2 - a_2 b_1}

The 3×3 Case

For three equations in three unknowns, compute four 3×3 determinants:

x = \frac{\det(A_1)}{\det(A)}, \quad y = \frac{\det(A_2)}{\det(A)}, \quad z = \frac{\det(A_3)}{\det(A)}

Each 3×3 determinant uses cofactor expansion. For matrix $\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$ :

\det = a(ei - fh) - b(di - fg) + c(dh - eg)

When to Use Cramer's Rule

Scenario	Best method
2×2 or 3×3 by hand	Cramer's rule (elegant, formulaic)
Need only one variable	Cramer's rule (compute just 2 determinants)
Large systems (4×4+)	Gaussian elimination (far fewer operations)
Nearly singular matrix	Gaussian elimination with pivoting

Computational cost: Cramer's rule for an $n \times n$ system requires computing $n+1$ determinants, each costing $O(n!)$ with cofactor expansion. For large $n$ , Gaussian elimination at $O(n^3)$ is dramatically faster.

When Cramer's Rule Fails

If $\det(A) = 0$ , Cramer's rule is undefined — division by zero. This means the system is either inconsistent (no solution) or dependent (infinitely many solutions). You must use row reduction to distinguish between these cases.

Frequently Asked Questions

What is Cramer's rule?

Cramer's rule expresses each unknown in a system of linear equations as a ratio of determinants: $x_i = \frac{\det(A_i)}{\det(A)}$ , where $A_i$ is the coefficient matrix $A$ with its $i$ th column replaced by the constant vector $\mathbf{b}$ .

When should I use Cramer's rule vs. Gaussian elimination?

Cramer's rule is best for 2x2 or 3x3 systems solved by hand, or when you need only one variable. For larger systems (4x4 and above), Gaussian elimination is dramatically faster because Cramer's rule requires computing $n+1$ determinants.

What happens when the determinant is zero?

When $\det(A) = 0$ , Cramer's rule is undefined (division by zero). This means the system has either no solution (inconsistent) or infinitely many solutions (dependent). You must use row reduction to distinguish between these two cases.

How do I compute a 3x3 determinant?

Use cofactor expansion along the first row: $\det = a(ei - fh) - b(di - fg) + c(dh - eg)$ , where the matrix entries are arranged as $(a,b,c;\\ d,e,f;\\ g,h,i)$ . Alternatively, use the Sarrus rule (diagonal products) for 3x3 matrices.

Can Cramer's rule find just one variable without solving the whole system?

Yes. To find only $x_i$ , you only need to compute two determinants: $\det(A)$ and $\det(A_i)$ . This is an advantage over Gaussian elimination, which always solves for all variables simultaneously.

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Cramer's Rule Calculator

Solve systems of linear equations using Cramer's Rule. See each determinant matrix step by step -- det(A), det(A_x), det(A_y) -- with highlighted column substitutions.

System Configuration

Equation 1

Equation 2

Try an Example

Simple 2x2 System

3x3 Engineering System

Parallel Lines (No Solution)

Cramer's Rule

The 2×2 Case

The 3×3 Case

When to Use Cramer's Rule

When Cramer's Rule Fails

Frequently Asked Questions

What is Cramer's rule?

When should I use Cramer's rule vs. Gaussian elimination?

What happens when the determinant is zero?

How do I compute a 3x3 determinant?

Can Cramer's rule find just one variable without solving the whole system?

Related method-focus Variants

gaussian

More Math Calculators

Cramer's Rule Calculator

Solve systems of linear equations using Cramer's Rule. See each determinant matrix step by step -- det(A), det(A_x), det(A_y) -- with highlighted column substitutions.

System Configuration

Equation 1

Equation 2

Try an Example

Simple 2x2 System

3x3 Engineering System

Parallel Lines (No Solution)

Cramer's Rule

The 2×2 Case

The 3×3 Case

When to Use Cramer's Rule

When Cramer's Rule Fails

Frequently Asked Questions

What is Cramer's rule?

When should I use Cramer's rule vs. Gaussian elimination?

What happens when the determinant is zero?

How do I compute a 3x3 determinant?

Can Cramer's rule find just one variable without solving the whole system?

Related method-focus Variants

gaussian

More Math Calculators

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