the vertices of a rectangle are given in the columns of the matrix $r = \\begin{bmatrix}0&0&3&3\\\\0&3&3&0\\e…

Answer:

D. (0, 0), (0, 3), (-3, 3), (-3, 0)

Explanation:

Step1: Recall matrix - multiplication rule

For a matrix - multiplication (A\times B), where (A) is an (m\times n) matrix and (B) is an (n\times p) matrix, the resulting matrix (C = A\times B) is an (m\times p) matrix. Here, (A=\begin{bmatrix}-1&0\0&1\end{bmatrix}) and (B = \begin{bmatrix}0&0&3&3\0&3&3&0\end{bmatrix}), and (C) will be a (2\times4) matrix.

Step2: Calculate the first - column of the product matrix

[ \begin{align*} \begin{bmatrix}-1&0\0&1\end{bmatrix}\times\begin{bmatrix}0\0\end{bmatrix}&=\begin{bmatrix}-1\times0 + 0\times0\0\times0+1\times0\end{bmatrix}=\begin{bmatrix}0\0\end{bmatrix} \end{align*} ]

Step3: Calculate the second - column of the product matrix

[ \begin{align*} \begin{bmatrix}-1&0\0&1\end{bmatrix}\times\begin{bmatrix}0\3\end{bmatrix}&=\begin{bmatrix}-1\times0+0\times3\0\times0 + 1\times3\end{bmatrix}=\begin{bmatrix}0\3\end{bmatrix} \end{align*} ]

Step4: Calculate the third - column of the product matrix

[ \begin{align*} \begin{bmatrix}-1&0\0&1\end{bmatrix}\times\begin{bmatrix}3\3\end{bmatrix}&=\begin{bmatrix}-1\times3+0\times3\0\times3 + 1\times3\end{bmatrix}=\begin{bmatrix}-3\3\end{bmatrix} \end{align*} ]

Step5: Calculate the fourth - column of the product matrix

[ \begin{align*} \begin{bmatrix}-1&0\0&1\end{bmatrix}\times\begin{bmatrix}3\0\end{bmatrix}&=\begin{bmatrix}-1\times3+0\times0\0\times3+1\times0\end{bmatrix}=\begin{bmatrix}-3\0\end{bmatrix} \end{align*} ] The columns of the resulting matrix give the coordinates of the transformed rectangle: ((0, 0)), ((0, 3)), ((-3, 3)), ((-3, 0)).