The problem provides a matrix $A$ and asks to find the element $a_{12}$.

1. Problem Description

The problem provides a matrix $A$ and asks to find the element $a_{12}$.

2. Solution Steps

The element $a_{12}$ refers to the element in the first row and second column of the matrix $A$.

Given the matrix

$A = \begin{bmatrix} 1 & 0 & -2 \\ 3 & 2 & 1 \\ 4 & 0 & 3 \end{bmatrix}$

The element in the first row and second column is

0. Therefore, $a_{12} = 0$.

3. Final Answer

$a_{12} = 0$