# Can you make up any matrix permutation with switching rows, columns, and transposition?

If I have a matrix of an arbitrary size, can I transform it into any matrix of same size that uses same elements using only switching rows/columns, and transposition (or transposition but using the other diagonal)? I looked into this answer, but it only says that it's impossible without transposing the matrix.

• Please clarify (possibly with an example) what do you mean with a transposition. – Andreas Caranti Jan 4 '17 at 16:53
• @AndreasCaranti It's the matrix transpose. – Arthur Jan 4 '17 at 16:55
• @Arthur, OK, thanks, I though it was some swap of elements inside the matrix. – Andreas Caranti Jan 4 '17 at 16:56

## 1 Answer

It's still impossible to go from, for instance, $\left[\begin{smallmatrix}1&1\\0&0\end{smallmatrix}\right]$ to $\left[\begin{smallmatrix}1&0\\0&1\end{smallmatrix}\right]$, because your operations don't change whether your matrix is invertible. In fact, they don't change the determinant at all except perhaps a sign change.

More concretely, if two element are on the same row or column, no matter what combination of operations you do, they still share either row or column afterwards.

• What operations could we add to make a complete system of operations? – RomaValcer Jan 4 '17 at 17:08
• @RomaValcer I believe something as simple as swapping the first and second element in the first row is enough. – Arthur Jan 4 '17 at 17:31