Quadratic Equation Solver (ax² + bx + c = 0)
About Matrix Inverse Calculator
The Matrix Inverse ($A^{-1}$) Calculator computes the multiplicative inverse of square $N\times N$ matrices such that $A \times A^{-1} = I$ using Gauss-Jordan Row Elimination $([A|I] \rightarrow [I|A^{-1}])$ and the Adjugate/Cofactor Matrix formula ($A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$).
How to Use Matrix Inverse Calculator
Step 1
Set square matrix dimension (e.g. $3\times3$).
Step 2
Fill in numerical entries for Matrix A.
Step 3
Click "Calculate Inverse Matrix (A⁻¹)".
Step 4
Review step-by-step Gauss-Jordan row operations and copy output.
Practical Use Cases for Matrix Inverse Calculator
Solving Systems of Linear Equations ($x = A^{-1}b$)
Solve matrix equation systems by multiplying the inverted coefficient matrix by the constants vector.
Computer Graphics View-Matrix Unprojection & Ray-Casting
Invert $4\times4$ camera projection matrices to convert 2D screen mouse clicks into 3D world-space picking rays.
Input & Output Examples
Inverting a 2x2 Matrix with Adjugate Formula
A = `[[4, 7], [2, 6]]`
Det = `10` | Inverse A⁻¹ = `[[0.6, -0.7], [-0.2, 0.4]]` (Exact: `[[3/5, -7/10], [-1/5, 2/5]]`)
Key Features & Performance
- ✓Algorithms: Gauss-Jordan Elimination with row augmented identity matrix and Adjugate/Cofactor formula.
- ✓Dimension support: $2\times2, 3\times3, 4\times4, \dots 8\times8$ square matrices.
- ✓Exact Rational Mode: keeps entries as exact reduced fractions (e.g. $3/7$) throughout all row operations.
- ✓100% Client-Side memory execution.
- ✓1-Click Copy inverse in LaTeX, Python NumPy, and JSON.
Key Terminology & Definitions
Matrix Invertibility
A square matrix $A$ has an inverse $A^{-1}$ if and only if its determinant is non-zero ($\det(A) \neq 0$).
Adjugate Matrix ($\text{adj}(A)$)
The transpose of the cofactor matrix of $A$, used in the analytical matrix inverse formula $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$.