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Inverse of a Matrix

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.

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# Inverse of a Matrix

## Matrix Modular Inverse Calculator

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.

### How to calculate the inverse of an invertible matrix?

The inverse of a square 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:

$$M^{-1} = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}^{-1} = \frac{1}{\det(M)} \begin{bmatrix} d & -b \\ -c & a \\ \end{bmatrix} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \\ \end{bmatrix}$$

Example: $$M = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix} \Rightarrow M^{-1} = \frac{1}{\det(M)} \begin{bmatrix} 4 & -2 \\ -3 & 1 \\ \end{bmatrix} = -\frac{1}{2} \begin{bmatrix} 4 & -2 \\ -3 & 1 \\ \end{bmatrix}$$

It is essential that the determinant of the matrix to be inverted is not equal to zero for the matrix to be invertible.

### How to prove that a matrix is invertible?

A matrix is invertible if its determinant is non-zero (different from 0).

A non inversible matrix is called singular.

### How to check that a matrix is the inverse of another?

The multiplication of the matrix by its inverse must give the identity matrix. So the computation of $M . M^{-1} = I$.

### How to calculate the modular inverse of a matrix?

The principle is the same, but instead of calculating the determinant, calculate the modular inverse of the matrix determinant.

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