Solve the linear system using an inverse matrix.
-$$\begin{aligned}
x + 2 y + 3 z & = 1 \
x + y + z & = 10 \
x - 2 z & = 7
\end{aligned}$$
A) $( 31 , - 33,12 )$
B) $( 21 , - 32,12 )$
C) $( 49,67,28 )$
D) $( 1,0,0 )$

Question

Quizplus · Accepted Answer

To solve the system using an inverse matrix, we first write the system in matrix form $AX = B$, where $A$ is the coefficient matrix, $X$ is the variable matrix, and $B$ is the constant matrix. For the given system:$$A = \begin{pmatrix} 1 & 2 & 3 \ 1 & 1 & 1 \ 1 & 0 & -2 \end{pmatrix}, \quad X = \begin{pmatrix} x \ y \ z \end{pmatrix}, \quad B = \begin{pmatrix} 1 \ 10 \ 7 \end{pmatrix}.$$To find $X$, we compute $X = A^{-1}B$. Without showing the detailed calculations of finding the inverse of $A$ and multiplying it by $B$, the correct solution is $(31, -33, 12)$, which corresponds to choice A.

Solve the Linear System Using an Inverse Matrix x+2y+3z=1x+y+z=10x−2z=7\begin{aligned}x + 2 y + 3 z & = 1 \\x + y + z & = 10 \\x - 2 z & = 7\end{aligned}x+2y+3zx+y+zx−2z​=1=10=7​

Solve the Linear System Using an Inverse Matrix $\begin{aligned}x + 2 y + 3 z & = 1 \\x + y + z & = 10 \\x - 2 z & = 7\end{aligned}$