Is the following matrix invertible?
$$\begin{bmatrix} 1235 &2344 &1234 &1990\\ 2124 & 4123& 1990& 3026 \\ 1230 &1234 &9095 &1230\\ 1262 &2312& 2324 &3907 \end{bmatrix}$$
Clearly, its determinant is not zero and, hence, the matrix is invertible.
Is there a more elegant way to do this?
Is there a pattern among these entries?
Find the determinant. To make calculations easier, work modulo $2$! The diagonal is $1$'s, the rest are $0$'s. The determinant is odd, and therefore non-zero.