Determinant Calculator

Calculate the determinant of 2x2, 3x3, and 4x4 matrices with cofactor expansion steps.

Select Matrix Size & Enter Values

Result

Determinant
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Matrix Size 2 x 2
Determinant --
Matrix is Singular? --

Step-by-Step Solution

det(A) = ad - bc

Understanding Matrix Determinants

The determinant is a scalar value that can be computed from the elements of a square matrix. It provides fundamental information about the matrix, including whether the matrix is invertible, the scaling factor of the linear transformation it represents, and the volume of the parallelepiped formed by its column vectors.

A matrix with a determinant of zero is called singular and does not have an inverse. Non-zero determinants indicate the matrix is invertible (non-singular). The sign of the determinant reveals whether the linear transformation preserves or reverses orientation.

Determinant Formulas by Size

2 x 2 Matrix

The simplest determinant calculation using the cross-multiplication rule.

det = ad - bc

3 x 3 Matrix

Computed using cofactor expansion along any row or column.

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

4 x 4 Matrix

Uses recursive cofactor expansion, reducing to 3x3 determinants.

det = sum of a_ij * C_ij along row 1

Properties

Key properties: det(AB) = det(A)det(B), det(A^T) = det(A), row swap changes sign.

det(kA) = k^n * det(A)

Cofactor Expansion Method

Cofactor expansion (also called Laplace expansion) is the standard method for computing determinants. You can expand along any row or column. For each element, compute its cofactor -- the minor determinant multiplied by (-1)^(i+j). The determinant is the sum of all element-cofactor products along the chosen row or column.

The minor Mij is the determinant of the submatrix formed by deleting row i and column j. The cofactor Cij = (-1)i+j * Mij. For practical computation, choose the row or column with the most zeros to minimize calculations.

Properties of Determinants

  • Row/Column operations: Swapping two rows changes the sign of the determinant.
  • Scalar multiplication: Multiplying a row by k multiplies the determinant by k.
  • Addition: Adding a multiple of one row to another does not change the determinant.
  • Triangular matrices: The determinant is the product of diagonal entries.
  • Transpose: det(AT) = det(A).
  • Product rule: det(AB) = det(A) * det(B).

Applications

Determinants are used extensively in linear algebra and its applications: solving systems of linear equations (Cramer's Rule), computing matrix inverses, calculating areas and volumes, analyzing stability of systems in engineering, and testing linear independence of vectors. In computer graphics, determinants help determine orientation and perform transformations.