Matrix Inverse Calculator

How it works: Form the augmented matrix $[A \mid I]$ and apply row operations to reduce A to the identity matrix I. The right half becomes $A^{-1}$.

Three row operations allowed:
1. $R_i \leftrightarrow R_j$ — Swap two rows
2. $R_i \leftarrow k \cdot R_i$ — Multiply a row by scalar $k \neq 0$
3. $R_i \leftarrow R_i + k \cdot R_j$ — Add a multiple of one row to another

Key fact: $A^{-1}$ exists if and only if $\det(A) \neq 0$.