Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { l }
- 2 x + y + 3 z - w = 14 \
- x - 4 y + z - 4 w = 12 \
- 4 x + 3 y - 2 z + w = 4 \
3 x - y - z - 4 w = 3
\end{array}$$
A) $\{ ( - 2,1,2 , - 3 ) \}$
B) $\{ ( - 2 , - 1 , - 2,3 ) \}$
C) $\varnothing$
D) $\{ ( - 4,2,3 , - 5 ) \}$

Question

Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { l } 
- 2 x + y + 3 z - w = 14 \
- x - 4 y + z - 4 w = 12 \
- 4 x + 3 y - 2 z + w = 4 \
3 x - y - z - 4 w = 3
\end{array}$$
A) $\{ ( - 2,1,2 , - 3 ) \}$
B) $\{ ( - 2 , - 1 , - 2,3 ) \}$
C) $\varnothing$
D) $\{ ( - 4,2,3 , - 5 ) \}$

Quizplus · Accepted Answer

To solve the system using the inverse of the coefficient matrix, we first write the system in matrix form $AX = B$, where $A$ is the coefficient matrix, $X$ is the column matrix of variables $(x, y, z, w)^T$, and $B$ is the column matrix of constants on the right side of the equations. Then, we find the inverse of $A$, denoted as $A^{-1}$, and multiply both sides by $A^{-1}$ to get $X = A^{-1}B$. The correct solution is obtained by performing these matrix operations, which results in the set $\{(-2, 1, 2, -3)\}$.