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Home › Math & Conversions › Core Math & Algebra › Matrix Determinant Calculator 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 size n×n 2 × 2 3 × 3 4 × 4 5 × 5 6 × 6 Decimal places Supports real-valued entries Matrix A Example 2×2 Example 3×3 Example 4×4 (sparse) Clear matrix Tip: use Tab
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Shift+Tab to move between cells. Empty cells are treated as 0. Calculate determinant Clear results 2. Results Determinant summary Size det(A) Sign of det(A) |det(A)| Singularity Invertibility Row operations Numerical epsilon Interpretation The determinant is built from the product of pivots during elimination. Row swaps flip the sign, while adding multiples of one row to another leaves det(A) unchanged. Row-operation log (Gaussian elimination) Copy as plain text 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 – FAQ What does a determinant of zero tell me? + If \( \det(A) = 0 \), then the matrix is singular: it does not have an inverse, and its rows or columns are linearly dependent. Geometrically, the transformation collapses volume to a lower-dimensional subspace. Why do you use Gaussian elimination instead of cofactor expansion? + Cofactor expansion has factorial complexity and becomes impractical for matrices larger than roughly 4×4. Gaussian elimination and LU decomposition scale as \( O(n^3) \), which is much more efficient and numerically stable for larger matrices. How accurate is the determinant for ill-conditioned matrices? + For ill-conditioned matrices, small changes in the entries can lead to large changes in the determinant. The calculator uses floating-point arithmetic with partial pivoting and a tolerance ( epsilon ) to decide when a pivot is effectively zero, but very ill-conditioned problems should still be interpreted with care. Can I use this calculator to test if a matrix is invertible? + Yes. A matrix is invertible if and only if its determinant is non-zero (within numerical tolerance). The summary panel explicitly states whether the matrix is classified as singular or invertible. Which entries can I use in the matrix? + The calculator accepts real-valued entries: integers, decimals, and negative numbers. Complex determinants are not supported in this version, but you can often separate real and imaginary parts if needed for theoretical work. Core Math & Algebra tools Limit Integral Derivative Matrix Rank Jacobian Matrix Gradient, Divergence, and Curl Standard Deviation Scientific Notation Weighted Average Dot Product Matrix Determinant – you are here Related linear algebra topics Eigenvalues & Eigenvectors Gaussian Elimination System of Linear Equations

Shift+Tab to move between cells. Empty cells are treated as 0. Calculate determinant Clear results 2. Results Determinant summary Size det(A) Sign of det(A) |det(A)| Singularity Invertibility Row operations Numerical epsilon Interpretation The determinant is built from the product of pivots during elimination. Row swaps flip the sign, while adding multiples of one row to another leaves det(A) unchanged. Row-operation log (Gaussian elimination) Copy as plain text 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 – FAQ What does a determinant of zero tell me? + If \( \det(A) = 0 \), then the matrix is singular: it does not have an inverse, and its rows or columns are linearly dependent. Geometrically, the transformation collapses volume to a lower-dimensional subspace. Why do you use Gaussian elimination instead of cofactor expansion? + Cofactor expansion has factorial complexity and becomes impractical for matrices larger than roughly 4×4. Gaussian elimination and LU decomposition scale as \( O(n^3) \), which is much more efficient and numerically stable for larger matrices. How accurate is the determinant for ill-conditioned matrices? + For ill-conditioned matrices, small changes in the entries can lead to large changes in the determinant. The calculator uses floating-point arithmetic with partial pivoting and a tolerance ( epsilon ) to decide when a pivot is effectively zero, but very ill-conditioned problems should still be interpreted with care. Can I use this calculator to test if a matrix is invertible? + Yes. A matrix is invertible if and only if its determinant is non-zero (within numerical tolerance). The summary panel explicitly states whether the matrix is classified as singular or invertible. Which entries can I use in the matrix? + The calculator accepts real-valued entries: integers, decimals, and negative numbers. Complex determinants are not supported in this version, but you can often separate real and imaginary parts if needed for theoretical work. Core Math & Algebra tools Limit Integral Derivative Matrix Rank Jacobian Matrix Gradient, Divergence, and Curl Standard Deviation Scientific Notation Weighted Average Dot Product Matrix Determinant – you are here Related linear algebra topics Eigenvalues & Eigenvectors Gaussian Elimination System of Linear Equations

Calculators in Shift+Tab to move between cells. Empty cells are treated as 0. Calculate determinant Clear results 2. Results Determinant summary Size det(A) Sign of det(A) |det(A)| Singularity Invertibility Row operations Numerical epsilon Interpretation The determinant is built from the product of pivots during elimination. Row swaps flip the sign, while adding multiples of one row to another leaves det(A) unchanged. Row-operation log (Gaussian elimination) Copy as plain text 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 – FAQ What does a determinant of zero tell me? + If \( \det(A) = 0 \), then the matrix is singular: it does not have an inverse, and its rows or columns are linearly dependent. Geometrically, the transformation collapses volume to a lower-dimensional subspace. Why do you use Gaussian elimination instead of cofactor expansion? + Cofactor expansion has factorial complexity and becomes impractical for matrices larger than roughly 4×4. Gaussian elimination and LU decomposition scale as \( O(n^3) \), which is much more efficient and numerically stable for larger matrices. How accurate is the determinant for ill-conditioned matrices? + For ill-conditioned matrices, small changes in the entries can lead to large changes in the determinant. The calculator uses floating-point arithmetic with partial pivoting and a tolerance ( epsilon ) to decide when a pivot is effectively zero, but very ill-conditioned problems should still be interpreted with care. Can I use this calculator to test if a matrix is invertible? + Yes. A matrix is invertible if and only if its determinant is non-zero (within numerical tolerance). The summary panel explicitly states whether the matrix is classified as singular or invertible. Which entries can I use in the matrix? + The calculator accepts real-valued entries: integers, decimals, and negative numbers. Complex determinants are not supported in this version, but you can often separate real and imaginary parts if needed for theoretical work. Core Math & Algebra tools Limit Integral Derivative Matrix Rank Jacobian Matrix Gradient, Divergence, and Curl Standard Deviation Scientific Notation Weighted Average Dot Product Matrix Determinant – you are here Related linear algebra topics Eigenvalues & Eigenvectors Gaussian Elimination System of Linear Equations.

Matrix Determinant Calculator

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