Consider non singular matrices made up of same entries having same determinant, are they all equivalent through some sequence of row and column swap?

Question

Suppose I have the matrix,

$$\begin{bmatrix} a &b \\ c & d \end{bmatrix}$$

Now, consider another matrix made up of same elements.

$$\begin{bmatrix} d &c \\ b & a \end{bmatrix}$$

This one above is also built up of same elements and has same determinant. Here we can map between the matrix by a row swap followed by a column swap (order doesn't matter).

Is that generally true tho? Can non singular matrix built up of same elements which have same determinant be turned into each other by row / column swaps?

The motivation comes from some me trying to solve this question: Counting non singular matrices

Consider a 3×3 matrix and the a11 entry and a22 entry of it. They can't be put in the same row and same column by any row swap or column swap. — Soumik Mukherjee, May 05 '24 at 08:02
They are still not in the same row or column right, so if you take a 3×3 matrix A with 1s in a11 and a22 and 0 elsewhere and another 3×3 matrix B with 1s in a11 and a12 and 0 elsewhere then you can't get B from A by row/column swiping — Soumik Mukherjee, May 05 '24 at 08:07
Ok, my mistake. You are correct. But, how does your point relate to the question? I believe that if you do such a swap, then you will neccesarily change the determinant @SoumikMukherjee — Clemens Bartholdy, May 05 '24 at 08:08
@trystwithfreedom You won't as the determinant of both matrices is 0 — GBA, May 05 '24 at 08:12
@Math101 What if I strengthen to only consider those with non zero determinant — Clemens Bartholdy, May 05 '24 at 08:23
consider $\begin{pmatrix} 2 & 1.5 & 2\ 0 & 3 & 1\ 0 & 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} 2 & 3 & 1\ 0 & 1.5 & 1\ 0 & 0 & 2 \end{pmatrix}$ — Soumik Mukherjee, May 05 '24 at 08:34
I'll check in a bit. Have some homework to do. @SoumikMukherjee — Clemens Bartholdy, May 05 '24 at 08:42
What about $\pmatrix{a&b\cr c&d\cr}$ and $\pmatrix{d&b\cr c&a\cr}$? — Gerry Myerson, May 05 '24 at 10:39
In general it is false for example for the above counterexample. But I wonder if it holds for symmetric matrices.. — Exodd, May 05 '24 at 10:56
@SoumikMukherjee no, because for $b=c$ it is the result of a row and then a column switch — Exodd, May 05 '24 at 19:01

score 3 · Accepted Answer · answered May 06 '24 at 06:06

3

To give an answer that meets all the requirements, even the requirement suggested in the comments that both matrices be symmetric, $$ \pmatrix{4&3&2\cr3&9&0\cr2&0&0\cr}\ {\rm\ and\ }\ \pmatrix{9&2&3\cr2&4&0\cr3&0&0\cr} $$ are both symmetric, have the same entries, and the same nonzero determinant, but you can't get from one to the other by a sequence of row swaps and column swaps.

answered May 06 '24 at 06:06

Gerry Myerson

185,413

good, but I was thinking. Is it true when instead of numbers we have variables? Take two symmetric matrices of variables. If they have the same (polynomial) determinant, is it true that they are a row-column swapped version of each other? – Exodd May 06 '24 at 08:38

Consider non singular matrices made up of same entries having same determinant, are they all equivalent through some sequence of row and column swap?

1 Answers1