Tool to calculate the permanent of a matrix, a value similar to the determinant, associated to a square matrix M denoted per(M).

Permanent of a Matrix - dCode

Tag(s) : Matrix

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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.

The permanent is like the determinant of a matrix, but without the signs - (minus).

Automatic method: use the dCode calculator above.

Manual method:

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

__Example:__ Si $ M = \begin{bmatrix} 1 & 2\\3 & 4 \end{bmatrix} $, alors $ \operatorname{per}(M) = 1 \times 4 + 2 \times 3 = 10 $

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.

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Source : https://www.dcode.fr/matrix-permanent

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