Find the best solutions to your problems with the help of IDNLearn.com. Get accurate answers to your questions from our community of experts who are always ready to provide timely and relevant solutions.

As Tia solved the system of equations below, she transformed matrices at different steps during the process:

[tex]\[
\begin{array}{l}
-a-b+2c=7 \\
2a+b+c=2 \\
-3a+2b+3c=7
\end{array}
\][/tex]

She noted the following matrices:

[tex]\[
\left[\begin{array}{lll|c}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & -1 \\
0 & 0 & 1 & 3
\end{array}\right]
\quad
\left[\begin{array}{ccc|c}
-1 & -1 & 2 & 7 \\
2 & 1 & 1 & 2 \\
-3 & 2 & 3 & 7
\end{array}\right]
\quad
\left[\begin{array}{ccc|c}
1 & 1 & -2 & -7 \\
0 & 1 & -5 & -16 \\
0 & 0 & 22 & 66
\end{array}\right]
\quad
\left[\begin{array}{lll|c}
1 & 1 & 0 & -1 \\
0 & 1 & 0 & -1 \\
0 & 0 & 1 & 3
\end{array}\right]
\][/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. III, II, IV, I
D. IV, I, II, III


Sagot :

To solve the system of equations using Gaussian elimination, we need to perform row operations to convert the matrix into its reduced row echelon form (RREF). The matrices provided correspond to different stages during this process. We need to arrange them in the correct order to understand the solution steps sequentially. Here are the detailed steps:

1. Initial System (Matrix II):
[tex]\[ \begin{bmatrix} -1 & -1 & 2 & 7 \\ 2 & 1 & 1 & 2 \\ -3 & 2 & 3 & 7 \\ \end{bmatrix} \][/tex]
This matrix represents the original system of equations:
[tex]\[ \begin{cases} -a - b + 2c = 7 \\ 2a + b + c = 2 \\ -3a + 2b + 3c = 7 \end{cases} \][/tex]

2. First Row Operation (Matrix III):
[tex]\[ \begin{bmatrix} 1 & 1 & -2 & -7 \\ 0 & 1 & -5 & -16 \\ 0 & 0 & 22 & 66 \\ \end{bmatrix} \][/tex]
The operations performed here likely involve swapping rows and scaling to get a leading 1 in the top-left corner followed by eliminating the first column below the pivot.

3. Second Row Operation (Matrix IV):
[tex]\[ \begin{bmatrix} 1 & 1 & 0 & -1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 3 \\ \end{bmatrix} \][/tex]
Here, further row operations have been performed to achieve zeros below the leading 1s in the second and first columns, leading towards the RREF form.

4. Final Reduced Row Echelon Form (Matrix I):
[tex]\[ \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 3 \\ \end{bmatrix} \][/tex]
This is the final form where the system of equations is fully reduced:
[tex]\[ \begin{cases} a = 0 \\ b = -1 \\ c = 3 \end{cases} \][/tex]

Thus, the correct order of matrices from start to finish in terms of solving the system of equations is:

1. Matrix II
2. Matrix III
3. Matrix IV
4. Matrix I
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thank you for choosing IDNLearn.com. We’re here to provide reliable answers, so please visit us again for more solutions.