Find the matrix that corresponds to a dilation centered at the origin with scale factor $-3.$
The dilation centered at the origin with scale factor $-3$ takes $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ to $\begin{pmatrix} -3 \\ 0 \end{pmatrix},$ and $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ to $\begin{pmatrix} 0 \\ -3 \end{pmatrix},$ so the matrix is
\[\boxed{\begin{pmatrix} -3 & 0 \\ 0 & -3 \end{pmatrix}}.\]
\begin{pmatrix}-3&0\0&-3\end{pmatrix}
Level 2
Precalculus
Problem: Find the matrix that corresponds to a dilation centered at the origin with scale factor $-3.$
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the matrix for a dilation transformation, we need to understand the properties of dilation in linear algebra. Dilation is a transformation that scales objects in space. The scale factor determines how much the object is enlarged or reduced. A negative scale factor indicates that the object will not only be scaled but also reflected across the origin.

In this case, the scale factor is -3. This means that every point in the space will be moved three times further away from the origin, and the direction will be reversed due to the negative sign. 

The standard matrix for a dilation centered at the origin with a scale factor \( k \) is given by the diagonal matrix:

\[
\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}
\]

Substituting \( k = -3 \) into this matrix form, we replace \( k \) with -3:

\[
\begin{pmatrix} -3 & 0 \\ 0 & -3 \end{pmatrix}
\]

This matrix indicates that both the x-coordinates and y-coordinates of any point will be multiplied by -3. The first row corresponds to the transformation of the x-coordinate, and the second row corresponds to the transformation of the y-coordinate. 

Thus, the final answer is:

\[
\begin{pmatrix} -3 & 0 \\ 0 & -3 \end{pmatrix}
\]<|END_OF_LATENT|> The final answer is $\begin{pmatrix}-3&0\0&-3\end{pmatrix}$.