Problem Statement
If $A = \begin{pmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} -2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & -2 \end{pmatrix}$, find the product $AB$. Hence, solve the system of linear equations:
$\begin{aligned} x - y + 2z &= 1 \\ 2y - 3z &= 1 \\ 3x - 2y + 4z &= 2 \end{aligned}$
Verified Solution & Marking Scheme
Compute Matrix Product AB
Multiplying row by column yields $AB = I_3$. Therefore, $A^{-1} = B$.
Formulate System in Matrix Form
The system is $AX = C$ where $C = \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}$.
Solve for X = BC
$X = \begin{pmatrix} -2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & -2 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 0 \\ 5 \\ 3 \end{pmatrix}$