Permanent of a Matrix

The permanent of a square matrix $ M = a_{i,j} $ is defined by $$ \operatorname{per}(M)=\sum_{\sigma\in P_n}\prod_{i=1}^n a_{i,\sigma(i)} $$ with $ P_n $ the permutations of $ n $ elements.

Automatic method: use the dCode calculator above.

Manual method:

For a 2x2 matrix, the calculation of the permanent is: $$ \operatorname{per} \left( \begin{bmatrix} a & b\\c & d \end{bmatrix} \right) = ad + bc $$

For higher size matrix like 3x3, the operation is similar:

$$ \operatorname{per} \left( \begin{bmatrix} a & b & c\\d & e & f\\g & h & i \end{bmatrix} \right) = a \operatorname{per} \left( \begin{bmatrix} e & f\\h & i \end{bmatrix} \right) + b \operatorname{per} \left( \begin{bmatrix} d & f\\g & i \end{bmatrix} \right) + c \operatorname{per} \left( \begin{bmatrix} d & e\\g & h \end{bmatrix} \right) \\ = aei+afh+bfg+bdi+cdh+ceg $$

The idea is the same for higher order matrices.

For a 1x1 matrix, the permanent is the only item of the matrix.

As for the determinant of a matrix, the permanent of a non-square matrix is not defined.