Use Matrices and Gauss-Jordan Elimination to Solve Systems
Solve the system of equations using matrices. Use Gauss-Jordan elimination.
-$$\begin{aligned}
3 x + 5 y + 2 w & = - 12 \
2 x + 6 z - w & = - 5 \
- 2 y + 3 z - 3 w & = - 3 \
- x + 2 y + 4 z + w & = - 2
\end{aligned}$$
A) $\{ ( - 1 , - 3,0,3 ) \}$
B) $\{ ( 1 , - 3,0,3 ) \}$
C) $\{ ( - 1,3,0 , - 3 ) \}$
D) $\{ ( 1,3,0 , - 3 ) \}$

Question

Quizplus · Accepted Answer

To solve the system of equations using Gauss-Jordan elimination, we first write the augmented matrix for the system and then apply row operations to transform it into reduced row echelon form (RREF). The given system is:$$\begin{align*}3x + 5y + 0z + 2w &= -12 \2x + 0y + 6z - w &= -5 \0x - 2y + 3z - 3w &= -3 \-x + 2y + 4z + w &= -2\end{align*}$$The augmented matrix is:$$\begin{bmatrix}3 & 5 & 0 & 2 & | & -12 \2 & 0 & 6 & -1 & | & -5 \0 & -2 & 3 & -3 & | & -3 \-1 & 2 & 4 & 1 & | & -2\end{bmatrix}$$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 that column 0), we aim to get the matrix into the form where the left side is the identity matrix. The solution to the system corresponds to the values in the rightmost column of the final matrix.After performing the necessary row operations, we find that the solution to the system is $x = -1$, $y = -3$, $z = 0$, and $w = 3$, which corresponds to choice A: $\{(-1, -3, 0, 3)\}$.

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

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