Quadratic Equation Solver (ax² + bx + c = 0)
About Matrix Calculator
The Comprehensive Linear Algebra Matrix Calculator computes Matrix Determinants ($\det(A)$), Matrix Inverses ($A^{-1}$), Transpose ($A^T$), Rank, Trace, Eigenvalues, Eigenvectors, and LU / QR / Cholesky decompositions for $N\times M$ matrices with step-by-step Gaussian elimination.
How to Use Matrix Calculator
Step 1
Set matrix grid dimensions (e.g. $3\times3$).
Step 2
Type or paste numerical elements into the grid cells.
Step 3
Select desired operation (Determinant, Inverse, Transpose, or RREF).
Step 4
Review step-by-step Gaussian elimination proof and copy result in LaTeX or NumPy.
Practical Use Cases for Matrix Calculator
Solving Systems of Linear Equations ($Ax = b$)
Solve systems of up to $10\times10$ linear algebraic equations using Cramer's Rule and Gaussian Row Reduction.
3D Computer Graphics Transformations & Physics Simulation
Calculate $4\times4$ perspective projection, rotation, scale, and affine transform matrix inverses for OpenGL and game engines.
Input & Output Examples
Calculating Inverse of 3x3 Matrix
Matrix A: `[[1, 2, 3], [0, 1, 4], [5, 6, 0]]`
Determinant: `1` | Inverse A⁻¹: `[[-24, 18, 5], [20, -15, -4], [-5, 4, 1]]` | Trace: `2` | Rank: `3`
Key Features & Performance
- ✓Matrix Dimensions: supports dynamic matrix sizes from $1\times1$ up to $10\times10$ matrices.
- ✓Operations: Determinant, Inverse ($A^{-1}$), Transpose ($A^T$), Adjugate, Trace, Rank, Eigenvalues, Gaussian Row Echelon Form (RREF).
- ✓Step-by-step row reduction steps with exact fraction arithmetic.
- ✓100% Client-Side linear algebra engine.
- ✓1-Click Copy result matrix in JSON, LaTeX, and Python NumPy formats.
Key Terminology & Definitions
Determinant ($\det(A)$)
A scalar value representing the scaling factor of the transformation; a matrix is invertible if and only if its determinant is non-zero.
Reduced Row Echelon Form (RREF)
The canonical form of a matrix after full Gaussian elimination where leading entries are 1 and columns containing pivots are zeroed.