Let's solve the problem step-by-step to find which matrix expression represents the geometric translation of the given polygon 4 units to the right and 5 units down.
We start with the original vertices of the polygon given at:
[tex]\[ (-5, 3), (-1, 3), (1, 0), (-3, 0) \][/tex]
We represent these vertices in matrix form, where the first row contains the x-coordinates and the second row contains the y-coordinates:
[tex]\[ \begin{pmatrix} -5 & -1 & 1 & -3 \\ 3 & 3 & 0 & 0 \end{pmatrix} \][/tex]
Next, we need to apply the translation. The translation matrix to move each vertex 4 units to the right and 5 units down is:
[tex]\[ \begin{pmatrix} 4 & 4 & 4 & 4 \\ -5 & -5 & -5 & -5 \end{pmatrix} \][/tex]
We now add the translation matrix to the original vertex matrix. This operation simply adds corresponding elements from each matrix:
[tex]\[ \begin{pmatrix} -5 & -1 & 1 & -3 \\ 3 & 3 & 0 & 0 \end{pmatrix} + \begin{pmatrix} 4 & 4 & 4 & 4 \\ -5 & -5 & -5 & -5 \end{pmatrix} \][/tex]
Let's perform the addition element-wise:
1. For the x-coordinates:
[tex]\[ -5 + 4 = -1 \][/tex]
[tex]\[ -1 + 4 = 3 \][/tex]
[tex]\[ 1 + 4 = 5 \][/tex]
[tex]\[ -3 + 4 = 1 \][/tex]
2. For the y-coordinates:
[tex]\[ 3 + (-5) = -2 \][/tex]
[tex]\[ 3 + (-5) = -2 \][/tex]
[tex]\[ 0 + (-5) = -5 \][/tex]
[tex]\[ 0 + (-5) = -5 \][/tex]
Combining these results, the translated vertices are:
[tex]\[ \begin{pmatrix} -1 & 3 & 5 & 1 \\ -2 & -2 & -5 & -5 \end{pmatrix} \][/tex]
This matches the matrix:
[tex]\[ \begin{pmatrix} -1 & 3 & 5 & 1 \\ -2 & -2 & -5 & -5 \end{pmatrix} \][/tex]
So the correct matrix expression representing a geometric translation of the given polygon 4 units to the right and 5 units down is:
[tex]\[ \left[ \begin{array}{cccc} -5 & -1 & 1 & -3 \\ 3 & 3 & 0 & 0 \end{array} \right] + \left[ \begin{array}{cccc} 4 & 4 & 4 & 4 \\ -5 & -5 & -5 & -5 \end{array} \right] \][/tex]
Thus, the correct choice is:
[tex]\[ \left[ \begin{array}{cccc} -5 & -1 & 1 & -3 \\ 3 & 3 & 0 & 0 \end{array} \right] + \left[ \begin{array}{cccc} 4 & 4 & 4 & 4 \\ -5 & -5 & -5 & -5 \end{array} \right] \][/tex]
