Use Matrices and Gauss-Jordan Elimination to Solve Systems
Solve the system of equations using matrices. Use Gauss-Jordan elimination.
-$$\begin{aligned}
x + y - z + w & = - 5 \
3 x - y + 3 z - 2 w & = 7 \
- 2 x + 2 y + z - w & = 16 \
- x - 2 y - 3 z + 3 w & = - 22
\end{aligned}$$
A) $\{ ( - 2,3,4 , - 2 ) \}$
B) $\left\{ \left( - 2 , - 3,5 , \frac { 1 } { 2 } \right) \right\}$
C) $\{ ( 2 , - 3 , - 4 , - 2 ) \}$
D) $\left\{ \left( \frac { 1 } { 2 } , - \frac { 1 } { 3 } , - \frac { 1 } { 4 } , - \frac { 1 } { 2 } \right) \right\}$

Question

Quizplus · Accepted Answer

To solve the system of equations using matrices and Gauss-Jordan elimination, we first write the augmented matrix of the system and then apply row operations to transform it into reduced row echelon form (RREF). The given system of equations is:$$\begin{aligned}x + y - z + w & = - 5 \3x - y + 3z - 2w & = 7 \-2x + 2y + z - w & = 16 \-x - 2y - 3z + 3w & = -22\end{aligned}$$The corresponding augmented matrix is:$$\begin{bmatrix}1 & 1 & -1 & 1 & |-5 \3 & -1 & 3 & -2 & |7 \-2 & 2 & 1 & -1 & |16 \-1 & -2 & -3 & 3 & |-22\end{bmatrix}$$Applying Gauss-Jordan elimination steps (which are too lengthy to detail in this brief explanation but involve making the leading coefficient of each row a 1 and then making all other entries in each column 0, except for the leading 1), we aim to get the matrix in RREF. The solution to the system corresponds to the values in the last column of the RREF matrix, which represent the values of $x$, $y$, $z$, and $w$, respectively.After performing the necessary row operations, we find that the solution to the system is $x = -2$, $y = 3$, $z = 4$, and $w = -2$, which matches choice A: $\{(-2, 3, 4, -2)\}$.

Use Matrices and Gauss-Jordan Elimination to Solve Systems Solve the System

Use Matrices and Gauss-Jordan Elimination to Solve Systems
Solve the System