Challenge: Matrix Determinant

Exercise

The determinant of a $(3 \times 3)$ matrix equals the volume of a parallelepiped represented by the matrix. The following illustration shows a parallelepiped where the sides are constructed using the vectors $\bf{a}$, $\bf{b}$, and $\bf{c}$:

To calculate the parallelepiped volume, we can use the formula below.

$$\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix} \to Determinant \to a_1b_2c_3 - a_1c_2b_3 + b_1c_2a_3 - b_1a_2c_3 + c_1a_2c_3 - c_1b_2a_3$$ ...