$A = [a_{ij}]$ is a 3 × 3 matrix with $a_{ij} = |i - j|$. What is the sum of all its elements?

Explanation

1. 01

   Element by element, $A = \begin{bmatrix}0 & 1 & 2 \\ 1 & 0 & 1 \\ 2 & 1 & 0\end{bmatrix}$: the leading diagonal is 0, and the rest is 1 next to it and 2 in the corners.

2. 02

   The sum is $4 \times 1 + 2 \times 2 = 8$.