Let $\mathbf{P}$ and $\mathbf{D}$ be the following $3\times 3$ matrices: \begin{align*} &\mathbf{P} \text{ projects vectors onto $\begin{pmatrix} 1 \\ 2 \\ 4 \end{pmatrix}$},\\ &\mathbf{D} \text{ dilates vectors by a factor of $3.$}\\ \end{align*}Calculate the matrix \[\mathbf{D} \mathbf{P} \begin{pmatrix} 1 & 0 & 2 \\ 2 & 0 & 1\\ 4 & 0 & -1 \end{pmatrix}.\]
So I noticed that the first column of the matrix is (1;2;4), so I'm thinking maybe we don't need to know what matrix P is. And I feel like dilating by a factor of 3 would be the identity matrix times a scalar 3. I'm stuck on how to apply those, or even if I am right. Any help would be appreicated
