What is the relationship between the determinants of the square matrices of equal dimensions $\mathbf{A}$ and $\mathbf{B}$ where each element of $\mathbf{B}$ is equal to the corresponding element of $\mathbf{A}$, times some constant ($\alpha$) raised to the absolute value of the difference of the row and column numbers? E.g. the $3 \times 3$ case is:
$ \mathbf{A} = \begin{vmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\ a_{2,1} & a_{2,2} & a_{2,3} \\ a_{3,1} & a_{3,2} & a_{3,3} \end{vmatrix} \qquad\&\qquad\mathbf{B} = \begin{vmatrix} \alpha^0 a_{1,1} & \alpha^1 a_{1,2} & \alpha^2 a_{1,3} \\ \alpha^1 a_{2,1} & \alpha^0 a_{2,2} & \alpha^1 a_{2,3} \\ \alpha^2 a_{3,1} & \alpha^1 a_{3,2} & \alpha^0 a_{3,3} \end{vmatrix} $
I haven't been able to get beyond trivial answers that are directly obvious from the problem statement.