## Precalculus (6th Edition) Blitzer

a) The linear system can be written as $\left[ \begin{matrix} 1 & -1 & 1 \\ 0 & 2 & -1 \\ 2 & 3 & 0 \\ \end{matrix} \right]\left[ \begin{matrix} x \\ y \\ z \\ \end{matrix} \right]=\left[ \begin{matrix} 8 \\ -7 \\ 1 \\ \end{matrix} \right]$ b) the value of the inverse matrix is $X=\left\{ \left( 2,-1,5 \right) \right\}$
(a) Consider the given system of equations: \begin{align} & x-y+z=8 \\ & 2y-z=-7 \\ & 2x+3y=1 \end{align} The linear system can be written as: $AX=B$ $\left[ \begin{matrix} 1 & -1 & 1 \\ 0 & 2 & -1 \\ 2 & 3 & 0 \\ \end{matrix} \right]\left[ \begin{matrix} x \\ y \\ z \\ \end{matrix} \right]=\left[ \begin{matrix} 8 \\ -7 \\ 1 \\ \end{matrix} \right]$ Where, $A=\left[ \begin{matrix} 1 & -1 & 1 \\ 0 & 2 & -1 \\ 2 & 3 & 0 \\ \end{matrix} \right]$ X=\left[ \begin{align} & x \\ & y \\ & z \\ \end{align} \right] B=\left[ \begin{align} & 8 \\ & -7 \\ & 1 \\ \end{align} \right] \ (b) Consider the given system of equations: \begin{align} & x-y+z=8 \\ & 2y-z=-7 \\ & 2x+3y=1 \end{align} The linear system can be written as: $AX=B$ $\left[ \begin{matrix} 1 & -1 & 1 \\ 0 & 2 & -1 \\ 2 & 3 & 0 \\ \end{matrix} \right]\left[ \begin{matrix} x \\ y \\ z \\ \end{matrix} \right]=\left[ \begin{matrix} 8 \\ -7 \\ 1 \\ \end{matrix} \right]$ Where, $A=\left[ \begin{matrix} 1 & -1 & 1 \\ 0 & 2 & -1 \\ 2 & 3 & 0 \\ \end{matrix} \right]$ X=\left[ \begin{align} & x \\ & y \\ & z \\ \end{align} \right] B=\left[ \begin{align} & 8 \\ & -7 \\ & 1 \\ \end{align} \right] Now, consider the coefficient matrix $A=\left[ \begin{matrix} 1 & -1 & 1 \\ 0 & 2 & -1 \\ 2 & 3 & 0 \\ \end{matrix} \right]$ Use the inverse of the coefficient matrix ${{\left[ A \right]}^{-1}}=\left[ \begin{matrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \\ \end{matrix} \right]$ Now, to find the values of the provided system: We will use the formula $X={{A}^{-1}}B$ Where, ${{\left[ A \right]}^{-1}}=\left[ \begin{matrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \\ \end{matrix} \right]$ B=\left[ \begin{align} & 8 \\ & -7 \\ & 1 \\ \end{align} \right] Now, substitute the values in $X={{A}^{-1}}B$