The inverse of a square matrix \(A\), denoted as \(A^{-1}\), is the matrix that, when multiplied by \(A\), results in the identity matrix \(I\).

\(A A^{-1} = A^{-1} A = I\)

A matrix only has an inverse if its determinant is non-zero.

For a 2x2 Matrix:

The inverse is found using the adjugate formula:
\( A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \)

For a 3x3 (or larger) Matrix:

A common method is using Gauss-Jordan elimination. An augmented matrix \([A|I]\) is created, and row operations are performed until it is transformed into the form \([I|A^{-1}]\). This calculator demonstrates that process step-by-step.