PX153 - K3 - solving simultaneous equations

\begin{aligned} x_{1} + 2 x_{2} & = 7 \\ 2 x_{1} + 3 x_{2} & = 12 \end{aligned}

\begin{aligned} (\begin{array}{c} a_{1} & a_{2} \\ a_{3} & a_{4} \end{array}) \cdot (\begin{array}{c} x_{1} \\ x_{2} \end{array}) & = (\begin{array}{c} b_{1} \\ b_{2} \end{array}) \\ A \cdot \vec{x} & = \vec{b} \end{aligned}

\begin{aligned} A \vec{x} & = \vec{b} \\ A^{- 1} A \vec{x} & = A^{- 1} \vec{b} \\ I \vec{x} & = A^{- 1} \vec{b} \\ ∴ \vec{x} & = A^{- 1} \vec{b} \end{aligned}

A^{- 1} = (\begin{matrix} - a_{4} & a_{3} \\ a_{2} & - a_{1} \end{matrix})

(\begin{matrix} x_{1} \\ x_{2} \end{matrix}) = (\begin{matrix} - a_{4} & a_{3} \\ a_{2} & - a_{1} \end{matrix}) (\begin{matrix} b_{1} \\ b_{2} \end{matrix})