Immovable Elements of a Matrix

Any matrix $A$ with $N$ rows and $M$ columns occupies N \times M cells in computer memory, just as any matrix $B$ with $M$ rows and $N$ columns does. Rows are numbered from $0$ to N - 1, and columns from $0$ to M - 1. The matrix is stored in memory in row-major order, which for matrix $A$ means the sequence: A_{0,0}, A_{0,1}, $\dots$, A_{0, M - 1}, A_{1, 0}, A_{1, 1}, $\dots$, A_{N - 1, M - 1}, so that adjacent elements occupy adjacent memory cells.

Matrix $B$ is called the transpose of matrix $A$ if B_{i, j} = A_{j, i} for all valid $i$ and $j$. Clearly, if matrix $A$ is transposed, it can be placed in the position of matrix $A$ itself. Then transposition can be seen as a permutation of some elements of matrix $A$. It is also clear that not all elements actually move; some remain immovable. For example, the first element of the first row and the last element of the last row remain immovable after transposition of any matrix.