Without expanding directly, prove that: $$\begin{vmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \end{vmatrix} = (a - b)(b - c)(c - a)$$

Given matrices $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix}$: (a) Compute $AB$ and $BA$, and show that $AB \ne BA$. (b) Verify that $(AB)^T = B^T A^T$.

Without expanding directly at first step, use properties of determinants to prove that: $$\begin{vmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \end{vmatrix} = (a - b)(b - c)(c - a)$$

Solve the following system of linear equations using the Matrix Inversion Method: $$2x + 3y = 8$$ $$5x - 2y = 1$$