Inverse Matrix Calculator
Inverse Matrix Calculator
Calculate the inverse of a square matrix using Gauss-Jordan elimination
Enter the matrix elements row by row. The matrix must be square (same number of rows and columns).
Inverse Matrix Calculator: Complete Linear Algebra Guide
The inverse of a matrix A is a matrix A⁻¹ such that AA⁻¹ = A⁻¹A = I (identity matrix).Matrix inversion is fundamental in linear algebra for solving systems of equations, computer graphics transformations, engineering calculations, and statistical analysis.
Our professional inverse matrix calculator uses Gauss-Jordan elimination to find the inverse of square matrices up to 5×5, providing step-by-step solutions, determinant calculation, and verification of results.
Quick Answer
To find matrix inverse: Use Gauss-Jordan elimination on the augmented matrix [A|I]. For a 2×2 matrix [[a,b],[c,d]], the inverse is (1/det) × [[d,-b],[-c,a]] where det = ad-bc. A matrix is invertible only if its determinant is non-zero.
Mathematical Foundation
Matrix multiplication property where A is the original matrix, A⁻¹ is its inverse, and I is the identity matrix
Key Concepts:
Determinant
A scalar value that determines if a matrix is invertible. If det(A) = 0, the matrix is singular and has no inverse. The determinant also represents the scaling factor of transformations.
Gauss-Jordan Elimination
Systematic method to find matrix inverse by transforming [A|I] into [I|A⁻¹] using elementary row operations: row swapping, scaling, and row addition/subtraction.
Identity Matrix
Square matrix with 1s on the diagonal and 0s elsewhere. Acts as the multiplicative identity for matrices: AI = IA = A for any compatible matrix A.
Matrix Inversion Methods
Gauss-Jordan Elimination
Most reliable method for computer implementation and education.
1. Form augmented matrix [A|I]
2. Use row operations to get [I|A⁻¹]
3. Right side becomes the inverse matrix2×2 Matrix Formula
Direct formula for 2×2 matrices only.
For A = [[a,b],[c,d]]
A⁻¹ = (1/(ad-bc)) × [[d,-b],[-c,a]]
Only if det = ad-bc ≠ 0Adjugate Method
Uses cofactor matrix and adjugate matrix.
A⁻¹ = (1/det(A)) × adj(A)
adj(A) = transpose of cofactor matrix
Cofactor = (-1)^(i+j) × minorApplications of Matrix Inversion
Engineering & Physics
Electrical Circuits
Solve circuit equations, impedance calculations, network analysis using nodal and mesh methods
Structural Analysis
Finite element analysis, stress-strain relationships, structural dynamics and vibration
Control Systems
State-space analysis, feedback control design, system stability and response
Computer Science
Computer Graphics
3D transformations, perspective projections, animation matrices, camera transformations
Machine Learning
Normal equations in linear regression, covariance matrix operations, PCA transformations
Cryptography
Hill cipher encryption/decryption, key generation in matrix-based cryptographic systems
Example Problems with Solutions
Example 1: 2×2 Matrix Inversion
Find the inverse of A = [[2, 1], [1, 1]]
Answer: A⁻¹ = [[1, -1], [-1, 2]]
Example 2: Solving Linear System
Solve: 2x + y = 5, x + y = 3 using matrix inversion
Answer: x = 2, y = 1
Example 3: Non-Invertible Matrix
Check if A = [[1, 2], [2, 4]] is invertible
Answer: Matrix is not invertible (determinant = 0)
Properties of Matrix Inverses
Basic Properties
Existence Conditions
Related Mathematical Tools
Inverse Matrix Calculator
Calculate the inverse of a square matrix using Gauss-Jordan elimination
Enter the matrix elements row by row. The matrix must be square (same number of rows and columns).