Evaluate $\begin{vmatrix}2 & -3 & 1 \\ 4 & 0 & -2 \\ 1 & 5 & 3\end{vmatrix}$.

Explanation

1. 01

   Expand along the first row, with the signs $+, -, +$.

2. 02

   The minors are $\begin{vmatrix} 0 & -2 \\ 5 & 3 \end{vmatrix} = 10$, $\begin{vmatrix} 4 & -2 \\ 1 & 3 \end{vmatrix} = 14$ and $\begin{vmatrix} 4 & 0 \\ 1 & 5 \end{vmatrix} = 20$.

3. 03

   $2(10) - (-3)(14) + 1(20) = 82$.