Cramer's Law Determinant Calculator

Results:
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ΔY =
ΔZ =
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What is a Cramer’s Rule Determinant Calculator?

A Cramer’s Rule Determinant Calculator is a tool used to solve systems of linear equations using Cramer’s Rule, which relies on determinants. This method provides an explicit formula for the solution of a system of N equations with N unknowns, given that the coefficient matrix has a nonzero determinant.

Why is Cramer’s Rule Important?

Cramer’s Rule is useful for:

  1. Solving Small Linear Systems (2×2, 3×3, etc.)
    • Directly finds the solution without row reduction or matrix inversion.
  2. Checking System Consistency
    • If det(A) = 0, the system has no unique solution (either dependent or inconsistent).
  3. Understanding Determinants' Role in Linear Algebra
    • Highlights the connection between determinants and solutions.
  4. Theoretical Applications in Physics & Engineering
    • Used in electrical circuits, statics, and structural analysis.

How Does Cramer’s Rule Work?

Consider a system of N linear equations with N unknowns:

Ax=b

where:

  • A is the coefficient matrix
  • x is the column vector of unknowns
  • b is the constant column vector

Step 1: Compute the Determinant of Matrix A

D=det⁡(A)

If D≠0 , the system has a unique solution.

Step 2: Compute Determinants for Modified Matrices

For each variable xi ​ , replace the i-th column of A with vector b and compute the determinant:

where Ai is the matrix A with its i-th column replaced by b.

Step 3: Compute Each Variable

Example: Solving a 2×2 System Using Cramer’s Rule

Solve:

Step 1: Construct Coefficient Matrix A

Compute det(A):

D=(2×(−1))−(3×4)=−2−12=−14

Step 2: Replace Columns and Compute Determinants

For x, replace the 1st column with b:

For y, replace the 2nd column with b:

Step 3: Compute Solutions

Final solution:

When to Use a Cramer’s Rule Determinant Calculator?

  • When solving small systems (2×2, 3×3, or 4×4) of linear equations.
  • When verifying manual solutions for linear equations.
  • When determining if a system has a unique solution (det(A) ≠ 0) or no unique solution (det(A) = 0).
  • When working with physics, engineering, or economics problems that involve linear relationships.