Quadratic Equation Solver (ax² + bx + c = 0)
About Matrix Determinant Calculator
The Matrix Determinant ($\det(A)$ or $|A|$) Calculator computes determinants for $2\times2$, $3\times3$, $4\times4$, and up to $10\times10$ square matrices using Laplace Cofactor Expansion, Sarrus Rule ($3\times3$), and Gaussian elimination with step-by-step arithmetic proofs.
How to Use Matrix Determinant Calculator
Step 1
Select square matrix size ($2\times2, 3\times3, 4\times4, \dots$).
Step 2
Enter numerical entries into matrix grid cells.
Step 3
Choose expansion method (Cofactor Expansion or Triangular Row Reduction).
Step 4
Review step-by-step diagonal multiplications and copy the exact determinant.
Practical Use Cases for Matrix Determinant Calculator
Determining Matrix Invertibility & Linear Independence
Check if a square matrix is non-singular (invertible) by verifying that its determinant is strictly non-zero ($\det(A) \neq 0$).
Calculating Geometric Scaling Factors & N-Dimensional Volumes
Compute the scaling factor of linear volume transformations (area in 2D, volume in 3D parallelepipeds) via matrix determinants.
Input & Output Examples
Calculating Determinant of 3x3 Matrix via Sarrus Rule
Matrix: `[[6, 1, 1], [4, -2, 5], [2, 8, 7]]`
Determinant: `-306` | Diagonal Products Proof: `(6×(-2)×7 + 1×5×2 + 1×4×8) - (1×(-2)×2 + 6×5×8 + 1×4×7) = -42 - 264 = -306`
Key Features & Performance
- ✓Methods: Laplace Cofactor Expansion, Rule of Sarrus ($3\times3$), Gaussian Lower/Upper Triangular Reduction.
- ✓Supports arbitrary matrix dimensions from $2\times2$ up to $10\times10$.
- ✓Exact fraction arithmetic: displays rational fraction answers ($a/b$) without decimal rounding error.
- ✓100% Client-Side linear algebra engine.
- ✓1-Click Copy determinant solution & step-by-step proof.
Key Terminology & Definitions
Determinant ($\det(A)$)
A scalar value computed from the elements of a square matrix that encodes geometric scaling and matrix invertibility properties.
Cofactor Expansion
A recursive method of computing matrix determinants by expanding along rows or columns using signed minor sub-determinants.