What is the determinant of a matrix?

The determinant of a square matrix A is a scalar value, denoted det(A), that summarizes important properties of the linear transformation associated with A. It measures the scale factor for areas (in 2D) or volumes (in 3D) and is zero exactly when the matrix is singular (non-invertible).

How do you compute the determinant of a 2×2 and 3×3 matrix?

For a 2×2 matrix [[a, b], [c, d]] the determinant is ad − bc. For a 3×3 matrix, you can use the rule of Sarrus or cofactor expansion: det(A) = a11(a22a33 − a23a32) − a12(a21a33 − a23a31) + a13(a21a32 − a22a31). For larger matrices, row operations or LU decomposition are typically used.

What does it mean if the determinant is zero?

If det(A) = 0, the matrix is singular: it does not have an inverse, its rows or columns are linearly dependent, and the associated linear transformation squashes n-dimensional volume to a lower-dimensional subspace.

Can rounding errors affect determinant computations?

Yes. Determinants of larger matrices can be sensitive to numerical rounding, especially when the matrix is ill-conditioned. This calculator uses Gaussian elimination with partial pivoting and applies a tolerance (epsilon) when deciding if a pivot is effectively zero.

Which matrix sizes does this determinant calculator support?

The calculator supports square matrices from 2×2 up to 6×6. You can select the dimension, fill the entries, and compute the determinant with a breakdown of the row operations used.

Matrix Determinant Calculator – 2×2, 3×3 and n×n with Row Operations

This matrix determinant calculator computes det(A) for 2×2, 3×3 and general n×n matrices (up to 6×6). It uses Gaussian elimination with partial pivoting for robust results and shows a row-operation log so you can follow every step.

Use it to check theoretical exercises, verify hand computations, or quickly test whether a matrix is invertible (non-singular). Ideal for linear algebra, numerical methods, and applied mathematics courses.

Matrix determinant – definition and key formulas

For a square matrix \( A \in \mathbb{R}^{n \times n} \), the determinant, denoted \( \det(A) \) or \( |A| \), is a scalar that encodes important information about the linear transformation associated with \( A \).

2×2 and 3×3 determinants

For a 2×2 matrix

\[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \quad\Rightarrow\quad \det(A) = ad - bc. \]

For a 3×3 matrix \( A = (a_{ij}) \), a common formula is

\[ \det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}). \]

General n×n determinant via cofactor expansion

For larger matrices, the determinant can be defined recursively by cofactor expansion along any row or column:

\[ \det(A) = \sum_{j=1}^{n} (-1)^{i+j} a_{ij} \det(M_{ij}), \]

where \( M_{ij} \) is the minor matrix obtained by removing row \( i \) and column \( j \). However, this formula becomes expensive for large \( n \), which is why practical computation uses row-reduction methods such as Gaussian elimination or LU decomposition.

Determinant via row operations

This calculator computes \( \det(A) \) by transforming the matrix into an upper triangular form using Gaussian elimination with partial pivoting:

Swapping two rows multiplies the determinant by −1.
Multiplying a row by a scalar \( k \) multiplies the determinant by \( k \).
Adding a multiple of one row to another leaves the determinant unchanged.

After reducing \( A \) to an upper triangular matrix \( U \), the determinant is the product of the pivots on the diagonal, adjusted for any row swaps performed along the way.

Geometric and algebraic interpretation

In 2D, \( |\det(A)| \) is the area scaling factor applied by the linear transformation.
In 3D, \( |\det(A)| \) is the volume scaling factor.
\( \det(A) = 0 \) if and only if the rows (or columns) are linearly dependent, meaning \( A \) is singular and non-invertible.
The sign of the determinant indicates whether the transformation preserves orientation (positive) or reverses it (negative).

Matrix Determinant Calculator

1. Define the matrix

Matrix A

2. Results

Determinant summary

Interpretation

Row-operation log (Gaussian elimination)

Matrix determinant – definition and key formulas

2×2 and 3×3 determinants

General n×n determinant via cofactor expansion

Determinant via row operations

Geometric and algebraic interpretation

Matrix determinant – FAQ