Inverse Matrix Calculator
Calculate the inverse of a square matrix using the Gauss-Jordan elimination method.
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How Matrix Inversion Works
The inverse of a square matrix A is denoted by A⁻¹ and satisfies:
\[ A \times A^{-1} = A^{-1} \times A = I \]
where I is the identity matrix.
Methods for finding the inverse:
- Gauss-Jordan Elimination:
- Augment the matrix with the identity matrix
- Perform row operations to convert the original matrix to identity
- The right side becomes the inverse
- Adjugate Method:
- Calculate the matrix of cofactors
- Transpose to get the adjugate
- Divide by the determinant
Properties:
- Only square matrices can have inverses
- A matrix is invertible if and only if its determinant is non-zero
- (AB)⁻¹ = B⁻¹A⁻¹
- (A⁻¹)⁻¹ = A
- (A^T)⁻¹ = (A⁻¹)^T
Applications:
- Solving systems of linear equations
- Computer graphics transformations
- Cryptography
- Statistics and data analysis
- Quantum mechanics