Compute the determinant of a 2×2 or 3×3 matrix via cofactor expansion (Laplace expansion). See minors, cofactors, and a full step-by-step expansion breakdown.
Cofactor expansion, also known as Laplace expansion, is a method for computing the determinant of a square matrix by expanding along a row or column. For each element in the chosen row or column, you compute its minor (the determinant of the sub-matrix obtained by deleting that element's row and column), apply a sign based on position, and sum the results.
For a 2×2 matrix [[a,b],[c,d]], the determinant is simply ad − bc. For a 3×3 matrix, cofactor expansion along the first row gives: det = a₁₁·C₁₁ + a₁₂·C₁₂ + a₁₃·C₁₃, where each Cᵢⱼ is the cofactor (−1)^(i+j) times the corresponding minor.
This method is fundamental in linear algebra and is used to derive matrix inverses (via the adjugate), solve systems of equations (Cramer's rule), and compute eigenvalues. While computationally expensive for large matrices, cofactor expansion provides clear theoretical insight and is practical for hand calculations on small matrices. This calculator shows every minor, every cofactor, the signed expansion terms, and the final determinant.
Cofactor Expansion (Determinant) Calculator helps you solve cofactor expansion (determinant) problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter your inputs once and immediately inspect Determinant, Is Singular?, Expansion Terms to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
det(A) = Σ aᵢⱼ · Cᵢⱼ along row i, where Cᵢⱼ = (−1)^(i+j) · Mᵢⱼ and Mᵢⱼ = minor determinant.
Result: Determinant shown by the calculator
Using the preset "2 × 2", the calculator evaluates the cofactor expansion (determinant) setup, applies the selected algebra rules, and reports Determinant with supporting checks so you can verify each transformation.
This calculator takes the problem inputs and applies the relevant cofactor expansion (determinant) relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use Determinant, Is Singular?, Expansion Terms, Sign of det to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
The minor Mᵢⱼ is the determinant of the sub-matrix formed by deleting row i and column j from the original matrix. Use this as a practical reminder before finalizing the result.
The cofactor Cᵢⱼ = (−1)^(i+j) × Mᵢⱼ. It is the signed minor, where the sign follows a checkerboard pattern.
No — the determinant is the same regardless of which row or column you choose. Choosing a row/column with zeros makes the computation faster.
A zero determinant means the matrix is singular: it has no inverse, its rows (or columns) are linearly dependent, and the system Ax = b either has no solution or infinitely many. Keep this note short and outcome-focused for reuse.
No. It has O(n!) time complexity. For matrices larger than 4×4, LU decomposition or Gaussian elimination is much faster. This tool is designed for educational 2×2 and 3×3 cases.
The inverse can be computed as A⁻¹ = (1/det) × adj(A), where adj(A) is the adjugate matrix (transpose of the cofactor matrix). See the Cofactor Matrix calculator.