Rank of a symbolic matrix not matching with the size of non-vanishing minor.

I have a large, symbolic (in one variable, x1) sparse matrix $N_{56x56}$. Using the `rank' command, matrix N has a rank of 52. I want to find the largest non-vanishing minor of N. I tried evaluating the determinant of all combinations of size-52 minors and all of them turned out to be zero.

The combinations of 52x52 minors are extracted from N as follows.

M1 = N(1:52,1:52),

M2 = N(1:52,2:53),

M3 = N(1:52,3:54),

...........................

M5= N(1:52,5:56),

M6 = N(2:53,1:52),

..............................

As the rank of N is 52, I was expecting the determinants of at-least one of the above minors to be non-zero, but they are all zeros. I would deeply appreciate any insights into this. Please let me know if any further information is required.

Answer 1

[Balaji]

Hi Isaac,

As per my understanding, you have some doubts about the existence of a minor of size 52.

Your original matrix is a 56-sized matrix of rank 52.

So there are number of ways you can chose the rows and number of ways you can chose the columns. So a total of * = 134901944100 number of 52 sized minors. So you need to check the determinant of all these combinations of minors.

Hope this helps!

Thanks

Balaji