Matrix Determinant Calculator
Matrix determinant calculator for 2×2, 3×3 and general n×n matrices. Enter values, compute det(A), see row-operation steps, and check if a matrix is singular or invertible.
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Matrix Determinant Calculator
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.
1. Define the matrix
Matrix A
Tip: use Tab / Shift+Tab to move between cells. Empty cells are treated as 0.
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
For a 3×3 matrix \( A = (a_{ij}) \), a common formula is
General n×n determinant via cofactor expansion
For larger matrices, the determinant can be defined recursively by cofactor expansion along any row or column:
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 – FAQ
Formula (LaTeX) + variables + units
A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \quad\Rightarrow\quad \det(A) = ad - bc.
\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}).
\det(A) = \sum_{j=1}^{n} (-1)^{i+j} a_{ij} \det(M_{ij}),
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\[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \quad\Rightarrow\quad \det(A) = ad - bc. \]
\[ \det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}). \]
\[ \det(A) = \sum_{j=1}^{n} (-1)^{i+j} a_{ij} \det(M_{ij}), \]
- No variables provided in audit spec.
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Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.