Let's analyze how to solve this system of equations using matrix multiplication. We are given the following system of equations:
[tex]\[
\begin{array}{l}
x + y + z = 160 \\
x - 2y - z = -100 \\
2x + 3y + 2z = 360 \\
\end{array}
\][/tex]
This system can be expressed in matrix form as [tex]\( A \mathbf{x} = \mathbf{B} \)[/tex], where:
[tex]\[
A = \begin{bmatrix}
1 & 1 & 1 \\
1 & -2 & -1 \\
2 & 3 & 2
\end{bmatrix},
\quad \mathbf{x} = \begin{bmatrix}
x \\
y \\
z
\end{bmatrix},
\quad \text{and} \quad B = \begin{bmatrix}
160 \\
-100 \\
360
\end{bmatrix}
\][/tex]
Next, let's evaluate each option to find the one that represents the solution.
### Option 1:
[tex]\[
\left[\begin{array}{ccc}1 & 1 & 1 \\ 1 & -2 & -1 \\ 2 & 3 & 2\end{array}\right] \left[\begin{array}{c}160 \\ -100 \\ 360\end{array}\right]
\][/tex]
Multiplying the matrix [tex]\( A \)[/tex] by vector [tex]\( B \)[/tex] gives [tex]\(A \mathbf{B}\)[/tex]. This operation uniquely resolves to check if [tex]\( A \mathbf{B} \)[/tex] results in [tex]\( B \)[/tex] which makes the system true. Based on the analysis:
[tex]\[
A \begin{bmatrix} 160 \\ -100 \\ 360 \end{bmatrix} = \begin{bmatrix} 160 \\ -100 \\ 360 \end{bmatrix}
\][/tex]
Therefore, option 1 provides the correct matrix equation representing the solution.
### Option 2:
[tex]\[
\left[\begin{array}{ccc}0.5 & -0.5 & 0.5 \\ 2 & 0 & -1 \\ 3.5 & 0.5 & 1.5\end{array}\right] \left[\begin{array}{c}360 \\ -100 \\ 160\end{array}\right]
\][/tex]
Evaluating this multiplication, it does not result in [tex]\( \mathbf{B} = \begin{bmatrix} 160 \\ -100 \\ 360 \end{bmatrix} \)[/tex].
### Option 3:
[tex]\[
\left[\begin{array}{ccc}-0.5 & 0.5 & 0.5 \\ -2 & 0 & 1 \\ 3.5 & -0.5 & -1.5\end{array}\right] \left[\begin{array}{c}160 \\ -100 \\ 360\end{array}\right]
\][/tex]
Similarly, evaluating this multiplication does not result in [tex]\( \mathbf{B} \)[/tex].
### Option 4:
[tex]\[
\left[\begin{array}{ccc}-1 & -1 & -1 \\ -1 & 2 & 1 \\ -2 & -3 & -2\end{array}\right] \left[\begin{array}{c}360 \\ -100 \\ 160\end{array}\right]
\][/tex]
Likewise, evaluating this multiplication, does not result in [tex]\( \mathbf{B} \)[/tex].
### Conclusion:
The option that satisfies [tex]\( A \mathbf{x} = \mathbf{B} \)[/tex] and correctly represents the solution to the system of equations is:
[tex]\[
\left[\begin{array}{ccc}1 & 1 & 1 \\ 1 & -2 & -1 \\ 2 & 3 & 2\end{array}\right]\left[\begin{array}{c}160 \\ -100 \\ 360\end{array}\right]
\][/tex]
Thus, the correct answer is the first option.
