Matrix Determinant Calculator

Calculate the determinant of 2×2, 3×3, and 4×4 matrices with step-by-step solutions.

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Understanding Matrix Determinants

The determinant is a scalar value that can be computed from the elements of a square matrix. It encodes important properties of the linear transformation described by the matrix, including whether the matrix is invertible and how it scales areas or volumes.

2×2 Determinant

Formula: For matrix [[a, b], [c, d]]

det(A) = ad − bc

The 2×2 determinant represents the signed area of the parallelogram formed by the column vectors of the matrix.

3×3 Determinant (Sarrus' Rule)

Formula: For matrix [[a, b, c], [d, e, f], [g, h, i]]

det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)

This is computed via cofactor expansion along the first row. The 3×3 determinant represents the signed volume of the parallelepiped formed by the column vectors.

4×4 Determinant (Cofactor Expansion)

For larger matrices, the determinant is computed by expanding along any row or column using cofactors:

det(A) = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃ + a₁₄C₁₄

Where C₁ⱼ = (−1)^(1+j) × M₁ⱼ, and M₁ⱼ is the minor (determinant of the submatrix formed by deleting row 1 and column j).

Key Properties of Determinants

Invertibility

A matrix is invertible (non-singular) if and only if det(A) ≠ 0
If det(A) = 0, the matrix is singular and has no inverse
det(A⁻¹) = 1 / det(A)

Multiplicative Property

det(AB) = det(A) × det(B)
det(kA) = k^n × det(A) for an n×n matrix
det(Aᵀ) = det(A)

Row Operations

Swapping two rows negates the determinant
Multiplying a row by scalar k multiplies the determinant by k
Adding a multiple of one row to another does not change the determinant

Special Matrices

det(I) = 1 for the identity matrix
Triangular matrix: determinant equals the product of diagonal entries
Orthogonal matrix: det(A) = ±1

Applications of Determinants

Determinants have wide-ranging applications across mathematics, science, and engineering:

Solving systems of equations: Cramer's Rule uses determinants to express solutions of systems of linear equations
Area and volume: The absolute value of the determinant gives the scale factor of the transformation (area in 2D, volume in 3D)
Eigenvalues: Found by solving det(A − λI) = 0, the characteristic equation
Cross products: The cross product of vectors in 3D can be expressed using a 3×3 determinant
Change of variables: The Jacobian determinant appears in multivariable calculus when changing coordinates

References

The formulas and methods used in this calculator are based on standard linear algebra principles:

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This calculator computes matrix determinants using cofactor expansion. Results are for educational and informational purposes. While we strive for accuracy, please verify important calculations independently. Floating-point arithmetic may introduce small rounding differences for very large values.

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