Inverse Matrix Calculator
Calculate matrix inverse using Gauss-Jordan elimination with step-by-step solutions
Matrix Input
Only square matrices can have inverses
Enter Matrix Elements
Matrix Inverse Results
Enter values in the matrix to calculate its inverse
The matrix must be square and non-singular (determinant ≠ 0)
Example: 2×2 Matrix Inverse
Input Matrix
[3, 4]
Solution Steps
Step 1: Calculate determinant: det(A) = (2)(4) - (1)(3) = 8 - 3 = 5
Step 2: Since det(A) ≠ 0, matrix is invertible
Step 3: For 2×2 matrix: A⁻¹ = (1/det(A)) × [d, -b; -c, a]
Step 4: A⁻¹ = (1/5) × [4, -1; -3, 2]
[-0.6, 0.4]
Matrix Inverse Properties
Square Matrix
Only square matrices can have inverses
Same number of rows and columns
Non-singular
Determinant must be non-zero
If det(A) = 0, no inverse exists
Identity Property
A × A⁻¹ = I
Multiplication gives identity matrix
Key Formulas
2×2 Matrix Inverse
A⁻¹ = (1/det(A)) × [d, -b; -c, a]
Determinant (2×2)
det(A) = ad - bc
General Formula
A⁻¹ = (1/det(A)) × adj(A)
Identity Check
A × A⁻¹ = A⁻¹ × A = I
Understanding Matrix Inverse
What is a Matrix Inverse?
The inverse of a matrix A is another matrix A⁻¹ such that when multiplied together, they produce the identity matrix. It's analogous to finding the reciprocal of a number, where a × (1/a) = 1. For matrices, A × A⁻¹ = I (identity matrix).
Requirements for Inverse
- •Square matrix: Must have equal rows and columns
- •Non-singular: Determinant must be non-zero
- •Full rank: All rows/columns must be linearly independent
Applications
- •Solving systems of linear equations
- •Computer graphics transformations
- •Cryptography and data encoding
- •Engineering and physics calculations
- •Machine learning and statistics
Note: When a matrix doesn't have an inverse (singular matrix), you can use the pseudoinverse for approximate solutions in applications like least squares regression.
Calculation Methods
Gauss-Jordan Elimination:
- • Create augmented matrix [A | I]
- • Use row operations to get [I | A⁻¹]
- • More numerically stable
- • Works for any size matrix
Adjugate Method:
- • Calculate matrix of cofactors
- • Transpose to get adjugate
- • Divide by determinant
- • Formula: A⁻¹ = adj(A)/det(A)