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Cramer's Rule

For Ax = b with A n×n invertible: xⱼ = det(Aⱼ)/det(A), where Aⱼ is A with column j replaced by b. Elegant but O(n⁴) — use Gaussian elimination for large n.

Concept Fundamentals
xⱼ = det(Aⱼ)/det(A)
Formula
A with col j → b
Aⱼ
det(A) ≠ 0
Requires
O(n⁴)
Cost

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Works only when det(A) ≠ 0. n+1 determinants for n variables. Gaussian elimination is O(n³) and preferred.

Key quantities
xⱼ = det(Aⱼ)/det(A)
Formula
Key relation
A with col j → b
Aⱼ
Key relation
det(A) ≠ 0
Requires
Key relation
O(n⁴)
Cost
Key relation

Ready to run the numbers?

Why: Cramer's rule gives explicit formula for each variable. Theoretically important; numerically expensive.

How: Compute det(A). For each j, form Aⱼ by replacing column j of A with b. xⱼ = det(Aⱼ) / det(A).

Works only when det(A) ≠ 0.n+1 determinants for n variables.
Sources:Gilbert Strang, Linear AlgebraWolfram MathWorld

Run the calculator when you are ready.

Solve via Cramer's Rulexⱼ = det(Aⱼ)/det(A)

Cramer's Rule Calculator

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What is Cramer's Rule?

Cramer's rule is a method for solving a system of linear equations using determinants. For a system of n equations with n unknowns, the solution for each variable can be expressed as the ratio of two determinants.

For a system Ax = b, the solution for the i-th variable (x_i) is given by:

xi=det⁡(Ai)det⁡(A)x_i = \frac{\det(A_i)}{\det(A)}

where A_i is the matrix formed by replacing the i-th column of A with the column vector b.

How Cramer's Rule Works

Step 1: Calculate det(A)

First, calculate the determinant of the coefficient matrix A.

Step 2: Check if det(A) ≠ 0

If det(A) = 0, the system either has no solution or has infinitely many solutions, and Cramer's rule cannot be applied directly. If det(A) ≠ 0, the system has a unique solution, and we can proceed.

Step 3: Create Matrices A_i

For each variable x_i, create a new matrix A_i by replacing the i-th column of A with the constants vector b.

Step 4: Calculate det(A_i)

Calculate the determinant of each A_i matrix.

Step 5: Calculate x_i

For each variable x_i, the solution is:

xi=det⁡(Ai)det⁡(A)x_i = \frac{\det(A_i)}{\det(A)}

Example: 2×2 System

Consider the system of equations:

{3x+2y=54x−y=3\begin{cases} 3x + 2y = 5 \\ 4x - y = 3 \end{cases}

Step 1: Write the coefficient matrix A and constants vector b:

A=(324−1),b=(53)A = \begin{pmatrix} 3 & 2 \\ 4 & -1 \end{pmatrix}, \quad b = \begin{pmatrix} 5 \\ 3 \end{pmatrix}

Step 2: Calculate det(A):

det⁡(A)=3×(−1)−2×4=−3−8=−11\det(A) = 3 \times (-1) - 2 \times 4 = -3 - 8 = -11

Step 3: Create A_1 by replacing column 1 with b:

A1=(523−1)A_1 = \begin{pmatrix} 5 & 2 \\ 3 & -1 \end{pmatrix}

Step 4: Calculate det(A_1):

det⁡(A1)=5×(−1)−2×3=−5−6=−11\det(A_1) = 5 \times (-1) - 2 \times 3 = -5 - 6 = -11

Step 5: Calculate x:

x=det⁡(A1)det⁡(A)=−11−11=1x = \frac{\det(A_1)}{\det(A)} = \frac{-11}{-11} = 1

Step 6: Create A_2 by replacing column 2 with b:

A2=(3543)A_2 = \begin{pmatrix} 3 & 5 \\ 4 & 3 \end{pmatrix}

Step 7: Calculate det(A_2):

det⁡(A2)=3×3−5×4=9−20=−11\det(A_2) = 3 \times 3 - 5 \times 4 = 9 - 20 = -11

Step 8: Calculate y:

y=det⁡(A2)det⁡(A)=−11−11=1y = \frac{\det(A_2)}{\det(A)} = \frac{-11}{-11} = 1

Therefore, the solution is x = 1, y = 1.

Advantages and Limitations

Advantages

  • Direct formula for the solution
  • Elegant mathematical method
  • Clear connection to linear algebra concepts
  • Works well for small systems (2×2, 3×3)
  • Provides explicit expressions for each variable

Limitations

  • Computationally expensive for large systems
  • Requires calculating n+1 determinants for n variables
  • Not applicable when det(A) = 0
  • More prone to numerical errors than methods like Gaussian elimination
  • Less efficient than other methods for computational implementation

Applications of Cramer's Rule

Computer Graphics

Cramer's rule is used to solve systems of equations for finding intersection points, coordinate transformations, and curve fitting in computer graphics applications.

Circuit Analysis

In electrical engineering, Cramer's rule can be used to analyze circuits with multiple nodes by solving systems of equations derived from Kirchhoff's laws.

Economics

Cramer's rule is applied in economic models to solve for equilibrium prices and quantities in systems with multiple markets or goods.

Theoretical Derivations

In mathematics and physics, Cramer's rule is often used for theoretical derivations and proofs, where an explicit formula for the solution is needed.

Historical Context

Cramer's rule is named after the Swiss mathematician Gabriel Cramer (1704-1752), who published the method in his 1750 work "Introduction à l'analyse des lignes courbes algébriques."

Although the rule bears Cramer's name, similar methods were known earlier. The Chinese mathematician Seki Takakazu developed comparable techniques in the 17th century, and Colin Maclaurin also discovered a form of the rule independently.

Despite its computational limitations, Cramer's rule remains an important theoretical tool in linear algebra and provides valuable insights into the relationships between determinants and solutions of linear systems.

For educational and informational purposes only. Verify with a qualified professional.

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xⱼ = det(Aⱼ)/det(A)

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