Question

each column of the matrix ( t=\begin{bmatrix}0&0&3\\0&3&0end{bmatrix} ) represents a vertex of a triangle. if rosa scales the triangle by finding ( 3t ), what are the vertices of the scaled triangle?
( (3,3),(3,9),(9,3) )
( (0,0),(0,6),(6,0) )
( (3,3),(0,9),(9,0) )
( (0,0),(0,9),(9,0) )

Explanation:

Step1: Multiply the matrix by the scalar

To find \(3T\), we multiply each element of matrix \(T=

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

\) by \(3\).
Using the rule for scalar - matrix multiplication \(k

$$\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\end{bmatrix}$$

=

$$\begin{bmatrix}ka_{11}&ka_{12}&ka_{13}\\ka_{21}&ka_{22}&ka_{23}\end{bmatrix}$$

\), we have:
\(3

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

=

$$\begin{bmatrix}3\times0&3\times0&3\times3\\3\times0&3\times3&3\times0\end{bmatrix}$$

=

$$\begin{bmatrix}0&0&9\\0&9&0\end{bmatrix}$$

\)

Step2: Identify the vertices

The columns of the matrix \(3T\) represent the vertices of the triangle.
The first column \(

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

\) gives the vertex \((0,0)\), the second column \(

$$\begin{bmatrix}0\\9\end{bmatrix}$$

\) gives the vertex \((0,9)\), and the third column \(

$$\begin{bmatrix}9\\0\end{bmatrix}$$

\) gives the vertex \((9,0)\)

Answer:

\((0,0),(0,9),(9,0)\) (corresponding to the fourth option)