Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { l }
2 x + 8 y + 6 z = 20 \
4 x + 2 y - 2 z = - 2 \
3 x - y + z = 11
\end{array}$$
A) $\{ ( 2,1,6 ) \}$
B) $\varnothing$
C) $\{ ( 2 , - 1,4 ) \}$
D) $\{ ( - 1,2,1 ) \}$

Question

Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { l } 
2 x + 8 y + 6 z = 20 \
4 x + 2 y - 2 z = - 2 \
3 x - y + z = 11
\end{array}$$
A) $\{ ( 2,1,6 ) \}$
B) $\varnothing$
C) $\{ ( 2 , - 1,4 ) \}$
D) $\{ ( - 1,2,1 ) \}$

Quizplus · Accepted Answer

To solve the system using the inverse of the coefficient matrix, we first write the system in matrix form as $AX = B$, where $A$ is the coefficient matrix, $X$ is the column matrix of variables $(x, y, z)^T$, and $B$ is the column matrix of constants on the right side of the equations. For this system, $A = \left[ \begin{array}{ccc} 2 & 8 & 6 \ 4 & 2 & -2 \ 3 & -1 & 1 \end{array} \right]$, $X = \left[ \begin{array}{c} x \ y \ z \end{array} \right]$, and $B = \left[ \begin{array}{c} 20 \ -2 \ 11 \end{array} \right]$. The solution is found by calculating $X = A^{-1}B$. After finding the inverse of $A$ and multiplying it by $B$, the solution is $X = \left[ \begin{array}{c} 2 \ -1 \ 4 \end{array} \right]$, which corresponds to the set $\{(2, -1, 4)\}$.

Solve the System by Using the Inverse of the Coefficient 2x+8y+6z=204x+2y−2z=−23x−y+z=11\begin{array} { l } 2 x + 8 y + 6 z = 20 \\4 x + 2 y - 2 z = - 2 \\3 x - y + z = 11\end{array}2x+8y+6z=204x+2y−2z=−23x−y+z=11​

Solve the System by Using the Inverse of the Coefficient $\begin{array} { l } 2 x + 8 y + 6 z = 20 \\4 x + 2 y - 2 z = - 2 \\3 x - y + z = 11\end{array}$