To find the coordinates of the vertices of the enlarged star-shaped figure, we will use the scaling factor provided and apply it to each coordinate in the original matrix.
The original coordinates of the vertices are given by the matrix:
[tex]\[
\begin{pmatrix}
0 & 5 & -5 & -3 & 3 \\
4 & 0 & 0 & -4 & -4
\end{pmatrix}
\][/tex]
The scaling factor is [tex]\(1.5\)[/tex].
To enlarge the figure by the given factor, we multiply each coordinate by the scaling factor. Let's calculate it step by step:
### Step 1: Multiply each element of the first row by the scaling factor
- [tex]\(0 \times 1.5 = 0\)[/tex]
- [tex]\(5 \times 1.5 = 7.5\)[/tex]
- [tex]\(-5 \times 1.5 = -7.5\)[/tex]
- [tex]\(-3 \times 1.5 = -4.5\)[/tex]
- [tex]\(3 \times 1.5 = 4.5\)[/tex]
This gives us the first row of the enlarged matrix:
[tex]\[
\begin{pmatrix}
0 & 7.5 & -7.5 & -4.5 & 4.5
\end{pmatrix}
\][/tex]
### Step 2: Multiply each element of the second row by the scaling factor
- [tex]\(4 \times 1.5 = 6\)[/tex]
- [tex]\(0 \times 1.5 = 0\)[/tex]
- [tex]\(0 \times 1.5 = 0\)[/tex]
- [tex]\(-4 \times 1.5 = -6\)[/tex]
- [tex]\(-4 \times 1.5 = -6\)[/tex]
This gives us the second row of the enlarged matrix:
[tex]\[
\begin{pmatrix}
6 & 0 & 0 & -6 & -6
\end{pmatrix}
\][/tex]
### Step 3: Compile the enlarged coordinates matrix
Combining both rows, we get the enlarged coordinates matrix:
[tex]\[
\begin{pmatrix}
0 & 7.5 & -7.5 & -4.5 & 4.5 \\
6 & 0 & 0 & -6 & -6
\end{pmatrix}
\][/tex]
### Step 4: Match our result with the given options
The result we obtained matches option a:
[tex]\[
\begin{pmatrix}
0 & 7.5 & -7.5 & -4.5 & 4.5 \\
6 & 0 & 0 & -6 & -6
\end{pmatrix}
\][/tex]
Hence, the coordinates of the vertices of the enlargement are:
[tex]\[
\begin{pmatrix}
0 & 7.5 & -7.5 & -4.5 & 4.5 \\
6 & 0 & 0 & -6 & -6
\end{pmatrix}
\][/tex]
and this corresponds to option a.
