## Precalculus (6th Edition) Blitzer

(b) Consider the given system of equations, \begin{align} & w-x+2y=-3 \\ & x-y+z=4 \\ & -w+x-y+2z=2 \\ & -x+y-2z=-4 \end{align} The linear system can be written as: $AX=B$ Where, $A=\left[ \begin{matrix} 1 & -1 & 2 & 0 \\ 0 & 1 & -1 & 1 \\ -1 & 1 & -1 & 2 \\ 0 & -1 & 1 & -2 \\ \end{matrix} \right]$ X=\left[ \begin{align} & w \\ & x \\ & y \\ & z \\ \end{align} \right] B=\left[ \begin{align} & -3 \\ & 4 \\ & 2 \\ & -4 \\ \end{align} \right] Now, consider the coefficient matrix $A=\left[ \begin{matrix} 1 & -1 & 2 & 0 \\ 0 & 1 & -1 & 1 \\ -1 & 1 & -1 & 2 \\ 0 & -1 & 1 & -2 \\ \end{matrix} \right]$ Use the inverse of the coefficient matrix, to get, ${{\left[ A \right]}^{-1}}=\left[ \begin{matrix} 0 & 0 & -1 & -1 \\ 1 & 4 & 1 & 3 \\ 1 & 2 & 1 & 2 \\ 0 & -1 & 0 & -1 \\ \end{matrix} \right]$ Now, to get the value of the provided system we will use the formula $X={{A}^{-1}}B$ Where, ${{\left[ A \right]}^{-1}}=\left[ \begin{matrix} 0 & 0 & -1 & -1 \\ 1 & 4 & 1 & 3 \\ 1 & 2 & 1 & 2 \\ 0 & -1 & 0 & -1 \\ \end{matrix} \right]$ B=\left[ \begin{align} & -3 \\ & 4 \\ & 2 \\ & -4 \\ \end{align} \right]