Where to start? Determinant of $n\times n$ matrix $A$ where $a_{ij} = b+d(ni+j)$

Question

I'm having difficulty starting this question for my linear algebra course, where would I begin?

Consider an $n\times n$ matrix $A$ where $a_{ij} = b+d(ni+j)$ where $b$ and $d \ne 0$ are fixed real numbers. What can you say about the determinant of A in the two cases $n = 2$ and $n > 2$.

Answer 1

If you make $C_2-C_1\to C_2$ you get that the elements of the new column are all $$b+d(in+2)-(b+d(in+1))=1.$$

If you make $C_3-C_2\to C_3$ you get that the elements of the new column are all $$b+d(in+2)-(b+d(in+2))=1.$$

So, if $n\ge 3$ the determinant is zero. For $n=2$ just write the matrix and get the determinant.