To reflect the given triangle over the line [tex]\( y = -x \)[/tex], we will start by understanding the original triangle's vertices and then performing the necessary transformation.
The original vertices of the triangle are given by the coordinates:
[tex]\[
\left[
\begin{array}{ccc}
0 & 7 & -1 \\
0 & -3 & -4
\end{array}
\right]
\][/tex]
These vertices correspond to the points [tex]\( (0,0), (7,-3), \text{and} (-1,-4) \)[/tex].
Reflecting a point over the line [tex]\( y = -x \)[/tex] can be achieved by transforming any point [tex]\( (x, y) \)[/tex] to [tex]\( (-y, -x) \)[/tex]. This reflection can be represented by the transformation matrix:
[tex]\[
\begin{bmatrix}
0 & -1 \\
-1 & 0
\end{bmatrix}
\][/tex]
Let's apply this transformation to each of the vertices:
1. For the vertex at [tex]\( (0, 0) \)[/tex]:
[tex]\[
\begin{pmatrix}
0 & -1 \\
-1 & 0
\end{pmatrix}
\begin{pmatrix}
0 \\
0
\end{pmatrix}
=
\begin{pmatrix}
0 \\
0
\end{pmatrix}
\][/tex]
So, the reflected vertex is [tex]\( (0, 0) \)[/tex].
2. For the vertex at [tex]\( (7, -3) \)[/tex]:
[tex]\[
\begin{pmatrix}
0 & -1 \\
-1 & 0
\end{pmatrix}
\begin{pmatrix}
7 \\
-3
\end{pmatrix}
=
\begin{pmatrix}
3 \\
-7
\end{pmatrix}
\][/tex]
So, the reflected vertex is [tex]\( (3, -7) \)[/tex].
3. For the vertex at [tex]\( (-1, -4) \)[/tex]:
[tex]\[
\begin{pmatrix}
0 & -1 \\
-1 & 0
\end{pmatrix}
\begin{pmatrix}
-1 \\
-4
\end{pmatrix}
=
\begin{pmatrix}
4 \\
1
\end{pmatrix}
\][/tex]
So, the reflected vertex is [tex]\( (4, 1) \)[/tex].
Therefore, the coordinates of the reflected triangle are:
[tex]\[
\left[
\begin{array}{ccc}
0 & 3 & 4 \\
0 & -7 & 1
\end{array}
\right]
\][/tex]
Thus, after reflecting the triangle over the line [tex]\( y = -x \)[/tex], the new vertices of the triangle are [tex]\( (0, 0), (3, -7), \text{and} (4, 1) \)[/tex].
