To convert the given system of equations into reduced row echelon form (RREF), we must examine the provided matrices step by step.
The system of equations is:
- \( 2x + y - 4z = -28 \)
- \( -2x + y + 3z = 21 \)
- \( x + y - z = -12 \)
We can represent this system as an augmented matrix:
\[ \begin{bmatrix} 2 & 1 & -4 & | & -28 \ -2 & 1 & 3 & | & 21 \ 1 & 1 & -1 & | & -12 \ \end{bmatrix} \]
We aim to use row operations to reduce it to RREF.
Let's analyze each provided matrix to check if they correctly represent the RREF of the given system:
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Matrix A: \[ \begin{bmatrix} 1 & 0 & 0 & | & -7 \ 0 & 1 & -1/2 & | & -7/2 \ 0 & 0 & 1 & | & 3 \ \end{bmatrix} \] In RREF, there should be a "1" in each leading position, and every leading "1" should be the only nonzero entry in its column. This matrix meets the RREF criteria.
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Matrix B: \[ \begin{bmatrix} 1 & 0 & 0 & | & -7 \ 0 & 1 & 0 & | & -2 \ 0 & 0 & 1 & | & 3 \ \end{bmatrix} \] This matrix also fulfills the RREF conditions. Each leading entry is "1" and is the only non-zero number in its respective column.
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Matrix C: \[ \begin{bmatrix} 1 & 0 & -7/4 & | & -49/4 \ 0 & 1 & -1/2 & | & -7/2 \ 0 & 0 & 1 & | & 3 \ \end{bmatrix} \] This matrix is not in RREF since the first column should not have any other coefficients apart from "1" in the leading position.
Given this analysis, Matrices A and B correspond to valid RREF forms of the original system. However, Matrix B is more straightforward as it maintains the values directly applicable from the original equations without introducing fractions.
Conclusion: Both Matrix A and Matrix B are valid as they produce RREF. If you are specifically looking for the reduced form without fractions, Matrix B is preferable. Thus, Matrix B is more appropriate for the context given.
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