To determine which of the provided matrices is the correct reduced row echelon form (RREF) conversion of the given system of equations:
- \( 2x + y - 4z = -28 \)
- \( -2x + y + 3z = 21 \)
- \( x + y - z = -12 \)
we need to convert the system into a matrix and then perform Gaussian elimination to form its RREF.
The augmented matrix for the system is:
\[ \begin{bmatrix} 2 & 1 & -4 & | & -28 \ -2 & 1 & 3 & | & 21 \ 1 & 1 & -1 & | & -12 \end{bmatrix} \]
Now, let's perform row operations to simplify this matrix to RREF.
- Start with the original matrix:
\[ \begin{bmatrix} 2 & 1 & -4 & | & -28 \ -2 & 1 & 3 & | & 21 \ 1 & 1 & -1 & | & -12 \end{bmatrix} \]
- We can simplify the first row by dividing by 2:
\[ \begin{bmatrix} 1 & \frac{1}{2} & -2 & | & -14 \ -2 & 1 & 3 & | & 21 \ 1 & 1 & -1 & | & -12 \end{bmatrix} \]
- Next, we can add 2 times the first row to the second row and subtract the first row from the third row:
\[ \begin{bmatrix} 1 & \frac{1}{2} & -2 & | & -14 \ 0 & 2 & -1 & | & -7 \ 0 & \frac{1}{2} & 1 & | & 2 \end{bmatrix} \]
- We can simplify the second row by dividing by 2:
\[ \begin{bmatrix} 1 & \frac{1}{2} & -2 & | & -14 \ 0 & 1 & -\frac{1}{2} & | & -\frac{7}{2} \ 0 & \frac{1}{2} & 1 & | & 2 \end{bmatrix} \]
- Now we need to eliminate the \( \frac{1}{2} \) in the third row’s second column by subtracting \( \frac{1}{2} \) times the second row from the third row:
\[ \begin{bmatrix} 1 & \frac{1}{2} & -2 & | & -14 \ 0 & 1 & -\frac{1}{2} & | & -\frac{7}{2} \ 0 & 0 & 1 & | & 3 \end{bmatrix} \]
- Finally, we can eliminate the \( \frac{1}{2} \) in the first row second column, and use the third row to adjust the second row:
After completing these operations correctly, the matrix should look like this in RREF:
\[ \begin{bmatrix} 1 & 0 & 0 & | & -7 \ 0 & 1 & 0 & | & -2 \ 0 & 0 & 1 & | & 3 \end{bmatrix} \]
Now, we can analyze the given matrices. The matrix that matches a correctly row-reduced version (i.e., each leading 1 in its own row with zeros above and below) indicates it is consistent with the RREF of a system.
After comparing the final forms, the matrix:
\[ \begin{bmatrix} 1 & 0 & 0 & | & -7 \ 0 & 1 & 0 & | & -2 \ 0 & 0 & 1 & | & 3 \end{bmatrix} \]
is the RREF version we're looking for.
Since this matrix doesn’t appear exactly among the options presented, we may find an equivalent representation or numerical errors. However, the most similar answer among the choices would need to check for consistency with leading variables and augmented version matching consisting of reduced elements.
Validating each option against values derived from transformations indicated might align with:
Result:
\[ \begin{bmatrix} 1 & 0 & 0 & | & -7 \ 0 & 1 & 0 & | & -2 \ 0 & 0 & 1 & | & 3 \end{bmatrix} \]
If re-alignment occurs further clarifications of matrix structures recognized under row operations on provided matrices would validate as correct, though as stated those numericals should be scrutinized.
2 answers