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.

Inverse of a Matrix - dCode

Tag(s) : Matrix

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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.

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.

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

A non **inversible matrix** is called *singular*.

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

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

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