Given an $m \times n$ binary matrix, matrix, return the maximum number of rows where all values become identical after flipping any number of columns. Flipping a column means changing every $0$ to $1$ and every $1$ to $0$ in that column.

Constraints:

• $m ==$ matrix.length

• $n==$ matrix[i].length

• $1 \leq m, n \leq 50$

• matrix[i][j] is either $0$ or $1$.

The essence of this solution lies in recognizing identical row patterns regardless of column flips. Instead of storing row values directly, we create a pattern string by comparing each element of the row to the first element—marking it as "T" if it matches and "F" otherwise. This binary pattern standardizes row representation, ensuring that rows that can be made identical via column flips have the same pattern. We then store pattern frequencies in a map, and the highest frequency determines the maximum number of identical rows achievable after flipping columns.

Using the intuition above, we implement the algorithm as follows:

1. Initialize a map, frequencies, to store the frequency of each pattern.

2. Iterate through each row in the matrix:

   1. Construct a pattern string by comparing each element in the row with the first element:

      1. If the element matches the first element, append "T" to the pattern.

      2. Otherwise, append "F".

   2. Update the frequency of this pattern in frequencies.

3. Initialize a variable res with 0 to track the maximum frequency found.

4. For each entry in frequencies, update res to the larger value between itself and entry’s frequency.

5. Return res as the final result.

Let’s look at the following illustration to get a better understanding of the solution: