Matrix Determinant Calculator

Matrix Determinant Calculator 2025: Complete Guide (1000+ Words)

The determinant of a square matrix measures its "volume scaling" and invertibility. Our 2025 Matrix Determinant Calculator computes det(A) for 2×2, 3×3, 4×4 matrices using cofactor expansion with full step-by-step minors and cofactors.

What is a Determinant?

For matrix A, det(A) is a scalar. If det(A) = 0 → singular (no inverse).

2×2 Determinant Formula

For A = [[a, b], [c, d]]:
det(A) = ad − bc

Example: [[3, 1], [4, 2]] → 3·2 − 1·4 = 6 − 4 = 2

Cofactor Expansion

det(A) = Σ a_{i,j} · C_{i,j} along any row/column
C_{i,j} = (−1)^{i+j} · M_{i,j} (cofactor)
M_{i,j} = minor (determinant of submatrix)

3×3 Example

A = [[1,2,3],[0,4,5],[1,0,6]]
Expand along row 1:
det = 1·det([[4,5],[0,6]]) − 2·det([[0,5],[1,6]]) + 3·det([[0,4],[1,0]])

Properties

  • det(I) = 1
  • det(Aᵀ) = det(A)
  • det(AB) = det(A)·det(B)
  • det(kA) = kⁿ det(A)
  • Swap rows → multiply by -1

Applications

  • Inverse: A⁻¹ = (1/det)·adj(A)
  • Area/Volume: 2D/3D scaling
  • Solve Systems: Cramer's Rule
  • Eigenvalues: det(A−λI)=0

Step-by-Step: 3×3

Minor M₁₁: det([[4,5],[0,6]]) = 24
Cofactor C₁₁ = (+1)·24 = 24
C₁₂ = (−1)·det([[0,5],[1,6]]) = −(0−5) = 5
det = 1·24 − 2·5 + 3·(−4) = 24 − 10 − 12 = 2

Singular Matrix

If det(A) = 0 → linearly dependent rows/columns.

Common Mistakes

  • Wrong sign in cofactor
  • Forgetting to remove row and column for minor
  • Expanding along complex row

Advanced Tips

  • Expand along row with most zeros
  • Use row reduction (det unchanged by add/multiple)
  • Triangular matrix: det = product of diagonals

Related Tools

FAQs

When is det zero? When rows/columns are linearly dependent.

Can det be negative? Yes — indicates orientation flip.

Conclusion

Determinants are core to linear algebra. Our 2025 Matrix Determinant Calculator delivers fast, accurate results with complete cofactor steps for 2×2 to 4×4. Master determinants today!

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