I have the following 15x15 binary matrix with a specific form: $$\begin{bmatrix} 1&1&0&0&0&0&1&1&1&1&1&1&0&0&0 \\ 1&0&1&0&0&1&0&1&1&1&0&0&1&1&0 \\ 1&0&0&1&0&1&1&0&1&0&1&0&1&0&1 \\ 1&0&0&0&1&1&1&1&0&0&0&1&0&1&1 \\ 0&1&1&0&0&1&1&0&0&0&1&1&1&1&0 \\ 0&1&0&1&0&1&0&1&0&1&0&1&1&0&1 \\ 0&1&0&0&1&1&0&0&1&1&1&0&0&1&1 \\ 0&0&1&1&0&0&1&1&0&1&1&0&0&1&1 \\ 0&0&1&0&1&0&1&0&1&1&0&1&1&0&1 \\ 0&0&0&1&1&0&0&1&1&0&1&1&1&1&0 \\ 1&1&1&1&0&1&1&1&1&1&1&1&1&1&1 \\ 1&1&1&0&1&1&1&1&1&1&1&1&1&1&1 \\ 1&1&0&1&1&1&1&1&1&1&1&1&1&1&1 \\ 1&0&1&1&1&1&1&1&1&1&1&1&1&1&1 \\ 0&1&1&1&1&1&1&1&1&1&1&1&1&1&1 \\ \end{bmatrix}$$
And I would like to show this matrix is full rank. (Spoilers: I know it will be full rank.) I want to generalize that matrices with this specific form will always have full rank, and I hope that working through this example and studying the structure of this matrix will help me generalize.
I'm being vague with "specific form"--I don't want to get too into the weeds--but I've observed that the submatrix $A$ formed by the last 5 rows and all 15 columns is particularly nice. It is pretty clear visually that this submatrix is full rank. Moreover, if we take the submatrix consisting of the final 5 rows and first 5 columns, notice that these are the indicator vectors for all the ways to choose 4 elements from a 5-element subset.
Similarly, note that the submatrix $B$ consisting of the first 10 rows and first 5 columns has indicator vectors for all the ways to choose 2 elements from a 5-element subset as the rows. These observations are related to the "specific form".
I know there are plenty of algorithms out there for determining the rank of a matrix. But since I want to generalize, I'm wondering if there's a more intuitive reason why matrices with this form will reduce nicely. Is there some sort of linear algebra trick or nice manipulation that can get me to a reduced form? Would it help to make this matrix sparse first, or looking for a block decomposition of sorts?
