Question

the vertices of a rectangle are given in the columns of the matrix $r = \

$$\begin{bmatrix} 0 & 0 & 6 & 6 \\\\ 0 & 6 & 6 & 0 \\end{bmatrix}$$

$. if $3 \times \

$$\begin{bmatrix} 0 & 1 \\\\ -1 & 0 \\end{bmatrix}$$

\times r$ is found to perform a transformation, what are the coordinates of the transformed rectangle?\
\bigcirc \\ (0, 0), (0, 18), (18, 18), (18, 0)\
\bigcirc \\ (0, 0), (18, 0), (18, -18), (0, -18)\
\bigcirc \\ (0, 0), (-18, 0), (-18, -18), (0, -18)\
\bigcirc \\ (0, 0), (18, 0), (18, 18), (0, 18)

Explanation:

Step1: Matrix Multiplication

First, calculate \(3\times

$$\begin{bmatrix}0&1\\-1&0\end{bmatrix}$$

\times

$$\begin{bmatrix}0&0&6&6\\0&6&6&0\end{bmatrix}$$

\).
Let \(A = 3\times

$$\begin{bmatrix}0&1\\-1&0\end{bmatrix}$$

=

$$\begin{bmatrix}0&3\\-3&0\end{bmatrix}$$

\) and \(B=

$$\begin{bmatrix}0&0&6&6\\0&6&6&0\end{bmatrix}$$

\).
The matrix - multiplication formula for \(C = A\times B\) is \(c_{ij}=\sum_{k = 1}^{2}a_{ik}b_{kj}\).
For \(i = 1,j = 1\): \(c_{11}=0\times0 + 3\times0=0\).
For \(i = 1,j = 2\): \(c_{12}=0\times0+3\times6 = 18\).
For \(i = 1,j = 3\): \(c_{13}=0\times6+3\times6 = 18\).
For \(i = 1,j = 4\): \(c_{14}=0\times6+3\times0 = 0\).
For \(i = 2,j = 1\): \(c_{21}=-3\times0+0\times0 = 0\).
For \(i = 2,j = 2\): \(c_{22}=-3\times0+0\times6 = 0\).
For \(i = 2,j = 3\): \(c_{23}=-3\times6+0\times6=-18\).
For \(i = 2,j = 4\): \(c_{24}=-3\times6+0\times0=-18\).

Step2: Interpret the Result

The resulting matrix \(

$$\begin{bmatrix}0&18&18&0\\0&0&-18&-18\end{bmatrix}$$

\) represents the coordinates of the vertices of the rectangle.
The columns of the matrix give the \((x,y)\) - coordinates of the vertices. The vertices are \((0,0)\), \((18,0)\), \((18, - 18)\), \((0,-18)\).

Answer:

\((0,0),(18,0),(18, - 18),(0,-18)\) (corresponding to the second option in the multiple - choice list)