E. Permutation of Rows and Columns

You have been given a matrix 𝑎 of size 𝑛 by 𝑚, containing a permutation of integers from 1 to 𝑛⋅𝑚.

A permutation of 𝑛 integers is an array containing all numbers from 1 to 𝑛 exactly once. For example, the arrays [1], [2,1,3], [5,4,3,2,1] are permutations, while the arrays [1,1], [100], [1,2,4,5] are not.

A matrix contains a permutation if, when all its elements are written out, the resulting array is a permutation. Matrices [[1,2],[3,4]], [[1]], [[1,5,3],[2,6,4]] contain permutations, while matrices [[2]], [[1,1],[2,2]], [[1,2],[100,200]]do not.

You can perform one of the following two actions in one operation:

• choose columns 𝑐 and 𝑑 (1≤𝑐,𝑑≤𝑚, 𝑐≠𝑑) and swap these columns;
• choose rows 𝑐 and 𝑑 (1≤𝑐,𝑑≤𝑛, 𝑐≠𝑑) and swap these rows.

You can perform any number of operations.

You are given the original matrix 𝑎 and the matrix 𝑏. Your task is to determine whether it is possible to transform matrix 𝑎 into matrix 𝑏 using the given operations.