Determinant Calculator

Compute the determinant of any square matrix (up to 6×6) using cofactor expansion — with full step-by-step breakdown of each minor and sign pattern.

Matrix Setup

Size (n×n)

How Cofactor Expansion Works

Cofactor Expansion (Laplace Expansion): For an n×n matrix A, expanding along the first row:

$$\det(A) = \sum_{j=1}^{n} (-1)^{1+j} \cdot a_{1j} \cdot M_{1j}$$ where $M_{1j}$ is the minor obtained by deleting row 1 and column j. The sign pattern follows $(-1)^{i+j}$: $$\begin{pmatrix} + & - & + & \cdots \\ - & + & - & \cdots \\ + & - & + & \cdots \end{pmatrix}$$

Determinant: how it is calculated

The determinant of a square matrix can be found by cofactor expansion along any row or column: det(A) = Σ (−1)^(i+j) · a(i,j) · M(i,j), where M(i,j) is the minor obtained by deleting row i and column j. Swapping two rows flips the sign, multiplying a row by k multiplies the determinant by k, and adding a multiple of one row to another leaves it unchanged. A matrix is invertible exactly when its determinant is not zero.

Common questions

What does a zero determinant mean?

The matrix is singular: it has no inverse and its rows (and columns) are linearly dependent.

Can I find the determinant of a non-square matrix?

No. The determinant is defined only for square matrices.

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