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Permanent of a Matrix

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

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Permanent of a Matrix -

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# Permanent of a Matrix

## Matrix NxN Permanent Calculator

### What is a matrix permanent? (Definition)

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).

### How to calculate a matrix permanent?

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.

### How to compute the permanent of a matrix 1x1?

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

### How to compute the permanent of a non square matrix?

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

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