gamezani30
Answered

[tex]$
-3a + 2b + 3c = 7
$[/tex]

She noted the following matrices:
[tex]$
\left[\begin{array}{lll|c}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & -1 \\
0 & 0 & 1 & 3
\end{array}\right] \quad\left[\begin{array}{ccc|c}
-1 & -1 & 2 & 7 \\
2 & 1 & 1 & 2 \\
-3 & 2 & 3 & 7
\end{array}\right] \quad\left[\begin{array}{ccc|c}
1 & 1 & -2 & -7 \\
0 & 1 & -5 & -16 \\
0 & 0 & 22 & 66
\end{array}\right] \quad\left[\begin{array}{ccc|c}
1 & 1 & 0 & -1 \\
0 & 1 & 0 & -1 \\
0 & 0 & 1 & 3
\end{array}\right]
$[/tex]

In which order should the matrices be arranged when solving the system from start to finish?

A. I, IV, III, II
B. II, III, IV, I
C. II, IV, I, III
D. IV, II, III, I



Answer :

GinnyAnswer
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].