Matrix Inverse Calculator
Compute the inverse of a square matrix via LU decomposition with singularity detection.
Inputs
A⁻¹
3 -1 -5 2
Determinant
1.00000000
Step by step
Matrix A
= 2 1 5 3
det(A)
= 1
A⁻¹ (via LU with partial pivoting)
= 3 -1 -5 2
How it works
The inverse A⁻¹ of a square matrix A satisfies A·A⁻¹ = I. This calculator uses LU decomposition with partial pivoting for numerical stability. If the matrix is singular (determinant effectively zero given a tolerance of 1e-12), no inverse exists.
Formulas
Inverse Definition
A · A⁻¹ = A⁻¹ · A = I
- A
- Square matrix
- I
- Identity matrix
LU Method
Solve L·U·xᵢ = eᵢ for each column of A⁻¹
- L
- Lower triangular
- U
- Upper triangular
- eᵢ
- i-th standard basis vector
Frequently Asked Questions
When does a matrix have no inverse?
A matrix is singular (not invertible) when its determinant is zero, meaning its rows or columns are linearly dependent.
Why use LU instead of cofactor expansion?
LU decomposition is O(n³) and numerically stable with partial pivoting, while cofactor expansion is O(n!) and impractical for matrices larger than 4×4.
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