Determinant Calculator
Calculate the determinant of any square matrix. Enter your matrix below and get instant results with detailed step-by-step solutions.
Matrix A
How to Calculate Matrix Determinant
Square Matrices Only
Determinant is only defined for square matrices (n×n). Non-square matrices don't have determinants.
Cofactor Expansion
Expand along any row or column using minors and cofactors for systematic calculation.
LU Decomposition
Decompose matrix into lower and upper triangular matrices for efficient computation.
Row Operations
Use elementary row operations to transform matrix to triangular form.
Mathematical Theory & History
What is a Determinant?
The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, including whether it's invertible, the volume scaling factor of the linear transformation it represents, and the orientation of the transformation.
2×2 Formula: det(A) = ad - bc
General Formula: det(A) = Σ(-1)^(i+j) * a_ij * M_ij
Where M_ij is the minor of element a_ij
Historical Background
The concept of determinants emerged in the 17th century from the work of Gottfried Leibniz on systems of linear equations. The modern notation and systematic theory were developed by Gabriel Cramer in 1750, leading to Cramer's rule for solving linear systems.
The term "determinant" was coined by Carl Friedrich Gauss in 1801. Later, Augustin-Louis Cauchy formalized the theory and introduced the notation we use today, making determinants a cornerstone of linear algebra.
Properties of Determinants
Multiplicative Property
det(AB) = det(A) × det(B)
Transpose Property
det(AT) = det(A)
Inverse Property
det(A-1) = 1/det(A)
Scalar Property
det(kA) = kndet(A)
Real-World Applications
Linear Systems
Determining if systems have unique solutions using Cramer's rule
Geometry
Calculating areas, volumes, and orientations in coordinate geometry
Physics & Engineering
Analyzing stability, eigenvalue problems, and transformation properties
Computer Graphics
Determining if transformations preserve orientation and calculating volumes
Frequently Asked Questions
No, determinants are only defined for square matrices (n×n). Non-square matrices (m×n where m≠n) do not have determinants. This is a fundamental mathematical restriction.
When det(A) = 0, the matrix is singular (non-invertible). This means the matrix doesn't have an inverse, the rows/columns are linearly dependent, and any linear system Ax = b either has no solution or infinitely many solutions.
For 2×2 and 3×3 matrices, cofactor expansion is straightforward. For larger matrices (4×4 and above), LU decomposition or Gaussian elimination are more efficient as they avoid the exponential growth in calculations.
A matrix is invertible if and only if its determinant is non-zero. If det(A) ≠ 0, then A⁻¹ exists and det(A⁻¹) = 1/det(A). This is a fundamental theorem in linear algebra.
The absolute value of the determinant represents the scaling factor of area (2D) or volume (3D) under the linear transformation. The sign indicates orientation: positive preserves orientation, negative reverses it.
Row swapping changes the sign of the determinant. Multiplying a row by scalar k multiplies the determinant by k. Adding a multiple of one row to another doesn't change the determinant.
Cramer's rule solves linear systems Ax = b when det(A) ≠ 0. Each variable equals det(Aᵢ)/det(A), where Aᵢ is A with column i replaced by b. It's useful for small systems but inefficient for large ones.