To determine the row echelon form of the given matrix
[tex]\[
\left[\begin{array}{ccc}
2 & 4 & 6 \\
-4 & 7 & 3 \\
4 & -1 & 2
\end{array}\right],
\][/tex]
we need to perform a series of row operations to transform it into a form where each leading entry of a row is to the right of the leading entry of the row above it, and all entries below each leading entry are zero.
Given the matrix has already been provided in row echelon form:
[tex]\[
\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]
\][/tex]
We can see that the correct answer is not among the given options:
A. [tex]\(\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 1 & 5 \\ 0 & 0 & 1\end{array}\right]\)[/tex]
B. [tex]\(\left[\begin{array}{ccc}1 & 2 & 3 \\ 0 & 15 & 15 \\ 4 & -1 & 2\end{array}\right]\)[/tex]
C. [tex]\(\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right]\)[/tex]
D. [tex]\(\left[\begin{array}{ccc}1 & 2 & 3 \\ 0 & 1 & 1 \\ 0 & -9 & 10\end{array}\right]\)[/tex]
The correct row echelon form, which is not listed among the multiple-choice options, is indeed:
[tex]\[
\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right].
\][/tex]
