Permanent

Remarks

Example: for $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, the two permutations give $\operatorname{perm}(A) = 1 \cdot 4 + 2 \cdot 3 = 10$, while $\det A = -2$.

The permanent is not multiplicative, unlike the determinant. For $J = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$, $\operatorname{perm}(J) = 2$, but $J^2 = 2J$ has permanent $8 \ne 2 \cdot 2$.

The permanent is linear in each row and each column, and it does not change when rows or columns are permuted. If $A$ is the $0/1$ biadjacency matrix of a bipartite graph with $n$ vertices on each side, then $\operatorname{perm}(A)$ is the number of perfect matchings of the graph. The permanent of the $n \times n$ all-ones matrix is $n!$.