🔄 Inverse Matrix Calculator
Calculate the inverse matrix A⁻¹ such that A × A⁻¹ = I
How to Use This Calculator
Select Matrix Size
Choose 2×2, 3×3, or 4×4 for your square matrix.
Enter Matrix Elements
Input all elements of your square matrix.
Calculate
Click to compute A⁻¹ using the adjoint method: A⁻¹ = adj(A) / det(A).
Verify
Check that A × A⁻¹ equals the identity matrix to confirm correctness.
Formula
A⁻¹ = adj(A) / det(A)
Where adj(A) is the adjoint (transpose of cofactor matrix)
For 2×2 Matrix:
If A = [a b; c d], then:
A⁻¹ = (1/(ad - bc)) × [d -b; -c a]
Note: det(A) = ad - bc must be non-zero
Properties:
- A × A⁻¹ = A⁻¹ × A = I (identity matrix)
- (AB)⁻¹ = B⁻¹A⁻¹
- (A⁻¹)⁻¹ = A
- (Aᵀ)⁻¹ = (A⁻¹)ᵀ
- det(A⁻¹) = 1 / det(A)
Requirements:
Matrix must be square and non-singular (det(A) ≠ 0)
About Inverse Matrix Calculator
The Inverse Matrix Calculator finds the inverse A⁻¹ of a square matrix A such that A × A⁻¹ = I, where I is the identity matrix. The inverse exists only if the matrix is non-singular (determinant ≠ 0). This calculator uses the adjoint method, computing the inverse via A⁻¹ = adj(A) / det(A).
When to Use This Calculator
- Solving Systems: Solve Ax = b using x = A⁻¹b
- Matrix Operations: Compute matrix division
- Linear Algebra: Find inverse for matrix equations
- Transformations: Reverse linear transformations
- Cryptography: Matrix-based encryption/decryption
Why Use Our Calculator?
- ✅ Complete Solution: Shows inverse, determinant, and verification
- ✅ Verification: Confirms A × A⁻¹ = I
- ✅ Error Detection: Identifies singular matrices
- ✅ Educational: Helps understand inverse calculation
- ✅ Accurate: Precise calculations
- ✅ Free: No registration required
Key Concepts
- Inverse: Matrix A⁻¹ such that AA⁻¹ = A⁻¹A = I
- Non-singular: det(A) ≠ 0 (inverse exists)
- Singular: det(A) = 0 (no inverse)
- Adjoint Method: A⁻¹ = adj(A) / det(A)
- Unique: If inverse exists, it's unique
Applications
Solving Linear Systems: For Ax = b, if A is invertible, then x = A⁻¹b.
Matrix Division: B/A = BA⁻¹ (not commutative).
Frequently Asked Questions
What is an inverse matrix?
The inverse A⁻¹ of matrix A satisfies A × A⁻¹ = A⁻¹ × A = I (identity matrix). It "undoes" the transformation represented by A.
Why does my matrix have no inverse?
If det(A) = 0, the matrix is singular and has no inverse. This means the matrix has linearly dependent rows/columns or reduces to a matrix with a zero row.
How is inverse calculated?
This calculator uses the adjoint method: A⁻¹ = adj(A) / det(A). Other methods include Gauss-Jordan elimination (augmenting [A | I] to get [I | A⁻¹]) and LU decomposition.
Is the inverse unique?
Yes! If an inverse exists, it's unique. There cannot be two different matrices that both satisfy AA⁻¹ = I.
Can I invert a non-square matrix?
No, only square matrices can have inverses. However, non-square matrices can have left or right inverses under certain conditions.