Are the $n \times n$ matrices with entries $a_{ij} = i \cdot n + j$ not invertible?

Question

Consider the matrices $A$ of size $n \times n$ with entries $a_{ij} = i \cdot n + j$. For example,

$\begin{pmatrix} 1 \end{pmatrix}$

$\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$

$\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$

$\begin{pmatrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{pmatrix}$

For $n \geq 3$ these seem to be singular with determinants $1, -2, 0, 0, \dots$

Is that conjecture correct? If yes, what are proofs or insights drawn from this?

Answer 1

Hint: row 1 + row 3 = $2 \cdot$ row 2.