Inverse of a Matrix

==2024-12-03

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To invert, it should be invertible.

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Theory

A matrix $A$ is said to be invertible if it’s determinant is not 0 or it is not a square matrix.

$A_{m\times n}$ is invertible if:

1. $m=n$
2. $|A|\ne 0$

Then, the Inverse of the Matrix $A$ is defined as:

$$A^{-1}=\frac{adj(A)}{|A|}$$

where, $adj(A)$ = Adjoint of a Matrix $A$ = $(A^c{)}^T$ where, $A^c$ = Matrix of Cofactors = $[-1^{(i+j)}{A}_{ij}]$ where, Cofactor at i,j position = $[-1^{(i+j)}{A}_{ij}]$ and, $A_{ij}$ is the Minor of Matrix at i,j position.

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Implementation

To find the inverse of a given square matrix for large matrices, we use Partitioned Matrices.

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PTR