To solve the given system [tex]\( -3a + 2b + 3c = 7 \)[/tex] using the provided matrices, we need to determine the correct sequence of matrices that represent the steps from the initial augmented matrix to the final solution.
1. Initial System Representation:
The system of equations can be written as an augmented matrix:
[tex]\[
\begin{pmatrix}
-3 & 2 & 3 & | & 7 \\
\end{pmatrix}
\][/tex]
2. Matrix II:
This matrix should represent the initial system:
[tex]\[
\begin{pmatrix}
-1 & -1 & 2 & | & 7 \\
2 & 1 & 1 & | & 2 \\
-3 & 2 & 3 & | & 7 \\
\end{pmatrix}
\][/tex]
This is where we start solving the system.
3. Matrix III:
This matrix represents an intermediate step where Gaussian elimination is partially applied:
[tex]\[
\begin{pmatrix}
1 & 1 & -2 & | & -7 \\
0 & 1 & -5 & | & -16 \\
0 & 0 & 22 & | & 66 \\
\end{pmatrix}
\][/tex]
4. Matrix IV:
Continuing with Gaussian elimination, we obtain this matrix:
[tex]\[
\begin{pmatrix}
1 & 1 & 0 & | & -1 \\
0 & 1 & 0 & | & -1 \\
0 & 0 & 1 & | & 3 \\
\end{pmatrix}
\][/tex]
5. Matrix I:
Finally, we reach the identity matrix with the solution:
[tex]\[
\begin{pmatrix}
1 & 0 & 0 & | & 0 \\
0 & 1 & 0 & | & -1 \\
0 & 0 & 1 & | & 3 \\
\end{pmatrix}
\][/tex]
So, the correct order to arrange the matrices when solving the system from start to finish is:
[tex]\[ \text{Matrix } II, \text{Matrix } III, \text{Matrix } IV, \text{Matrix } I \][/tex]
Therefore, the order is [tex]\( II, III, IV, I \)[/tex].
