91Ó°ÊÓ

Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. $$\begin{aligned}w+2 x+3 y-z &=7 \\\2 x-3 y+z &=4 \\\w-4 x+y &=3\end{aligned}$$

Use Cramer's rule to solve each system or to determine that the system is inconsistent or contains dependent equations. $$ \begin{aligned}&3 x=2-3 y\\\&2 y=3-2 x\end{aligned} $$

In Exercises \(19-24\), perform each matrix row operation and write the new matrix. \(\left[\begin{array}{rrrr|r}1 & -1 & 1 & 1 & 3 \\ 0 & 1 & -2 & -1 & 0 \\ 2 & 0 & 3 & 4 & 11 \\ 5 & 1 & 2 & 4 & 6\end{array}\right] \begin{array}{r}-2 R_{1}+R_{3} \\ -5 R_{1}+R_{4}\end{array}\) \(\left[\begin{array}{rrrr|r}1 & -5 & 2 & -2 & 4 \\ 0 & 1 & -3 & -1 & 0 \\ 3 & 0 & 2 & -1 & 6 \\ -4 & 1 & 4 & 2 & -3\end{array}\right] \begin{array}{r}-3 R_{1}+R_{3} \\ 4 R_{1}+R_{4}\end{array}\)

In Exercises \(17-26,\) let $$ A=\left[\begin{array}{rr} -3 & -7 \\ 2 & -9 \\ 5 & 0 \end{array}\right] \text { and } B=\left[\begin{array}{rr} -5 & -1 \\ 0 & 0 \\ 3 & -4 \end{array}\right] $$ Solve each matrix equation for \(X\). $$ A-X=4 B $$

Use Cramer's rule to solve each system or to determine that the system is inconsistent or contains dependent equations. $$ \begin{aligned}&x+2 y-3=0\\\&12=8 y+4 x\end{aligned} $$

find \(A^{-1}\) by forming \([A | I]\) and then using row operations to obtain [ \(I | B],\) where \(A^{-1}=[B]\) Check that \(A A^{-1}=I\) and \(A^{-1} A=I\) $$ A=\left[\begin{array}{lll} 2 & 4 & -4 \\ 1 & 3 & -4 \\ 2 & 4 & -3 \end{array}\right] $$

In Exercises \(17-26,\) let $$ A=\left[\begin{array}{rr} -3 & -7 \\ 2 & -9 \\ 5 & 0 \end{array}\right] \text { and } B=\left[\begin{array}{rr} -5 & -1 \\ 0 & 0 \\ 3 & -4 \end{array}\right] $$ Solve each matrix equation for \(X\). $$ 4 A+3 B=-2 X $$

find \(A^{-1}\) by forming \([A | I]\) and then using row operations to obtain [ \(I | B],\) where \(A^{-1}=[B]\) Check that \(A A^{-1}=I\) and \(A^{-1} A=I\) $$ A=\left[\begin{array}{rrr} 5 & 0 & 2 \\ 2 & 2 & 1 \\ -3 & 1 & -1 \end{array}\right] $$

Use Cramer's rule to solve each system or to determine that the system is inconsistent or contains dependent equations. $$ \begin{aligned}&4 y=16-3 x\\\&6 x=32-8 y\end{aligned} $$

In Exercises \(17-26,\) let $$ A=\left[\begin{array}{rr} -3 & -7 \\ 2 & -9 \\ 5 & 0 \end{array}\right] \text { and } B=\left[\begin{array}{rr} -5 & -1 \\ 0 & 0 \\ 3 & -4 \end{array}\right] $$ Solve each matrix equation for \(X\). $$ 4 B+3 A=-2 X $$