Inverse of a Matrix

The inverse of a square matrix $ M $ is noted $ M^{-1} $ and can be calculated in several ways. The most suitable for 2x2 or 3x3 matrix sizes is the cofactor method which necessitate to calculate the determinant of the matrix $ \det M $ and the transposed cofactor matrix (also called adjugate matrix $ \operatorname{adj}(M) $):

$$ M^{-1} = \frac{1}{\det M} \left( \operatorname{cof}(M) \right)^\mathsf{T} = \frac{1}{\det M} \operatorname{adj}(M) $$

The dCode calculator works for any size of square matrix.

For a 2x2 matrix (order 2):

$$ M = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} \\ \det(M) = ad - bc \\ \operatorname{cof}(M) = \begin{bmatrix} d & -c \\ -b & a \end{bmatrix} \\ \operatorname{adj}(M) = \left( \operatorname{cof}(M) \right)^\mathsf{T} = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \\ M^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \\ \end{bmatrix} $$

For a 3x3 matrix (order 3):

$$ M^{-1} = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}^{-1} = \left( \begin{bmatrix} \frac{e i-f h}{-c e g+b f g+c d h-a f h-b d i+a e i} & \frac{c h-b i}{-c e g+b f g+c d h-a f h-b d i+a e i} & \frac{b f-c e}{-c e g+b f g+c d h-a f h-b d i+a e i} \\ \frac{f g-d i}{-c e g+b f g+c d h-a f h-b d i+a e i} & \frac{a i-c g}{-c e g+b f g+c d h-a f h-b d i+a e i} & \frac{c d-a f}{-c e g+b f g+c d h-a f h-b d i+a e i} \\ \frac{d h-e g}{-c e g+b f g+c d h-a f h-b d i+a e i} & \frac{b g-a h}{-c e g+b f g+c d h-a f h-b d i+a e i} & \frac{a e-b d}{-c e g+b f g+c d h-a f h-b d i+a e i} \end{bmatrix} \right) $$

A matrix is invertible if its determinant is non-zero (different from 0). So to prove that a matrix has an inverse, calculate the determinant of the matrix, if it is different from 0, then the matrix is invertible.

A matrix with a determinant equal to 0 is not invertible. It does not have an inverse, it is not possible to calculate its inverse.

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.