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) : Mathematics,Algebra,Symbolic Computation

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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 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:

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

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

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

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 one has to calculate the modular inverse of the matrix determinant.

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inverse,matrix,square,identity,inversion,invertible,nonsingular,nondegenerate

Source : http://www.dcode.fr/matrix-inverse

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