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Tool to invert a matrix. The inverse of a square matrix M is a matrix denoted M^{-1} such as que M.M^{-1}=I where I is the identity matrix.

Answers to Questions

How to calculate the inverse of an invertible matrix?

The inverse of a matrix is calculated in several ways, the easiest is the cofactor method which necessitate to calculate the determinant of the matrix but also the comatrix and its transposed matrix :$$ M^{-1}=\frac1{\det M} \,^{\operatorname t}\!{{\rm com} M} = \frac1{\det M} \,^{\rm t}\!C $$For a 2x2 matrix it gives :$$ \mathbf{M}^{-1} = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}^{-1} = \frac{1}{\det(\mathbf{M})} \begin{bmatrix} \,\,\,d & \!\!-b \\ -c & \,a \\ \end{bmatrix} = \frac{1}{ad - bc} \begin{bmatrix} \,\,\,d & \!\!-b \\ -c & \,a \\ \end{bmatrix}. $$

How to calculate the modular inverse of a matrix?

The principle is the same, but one has to calculate the modular inverse of the matrix determinant.

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