One particular example would be a matrix that has 0's everywhere except for one 1 in every column and every row. This is called a permutation matrix.

Heinrich Steinlein, Fermat′s Little Theorem and Gauss Congruence: Matrix Versions and Cyclic Permutations, The American Mathematical Monthly, Vol. 124, No. 6 (June-July 2017), pp. 548-553 ...