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 \(\mathbf{D} \mathbf{P} \begin{pmatrix} 1 & 0 & 2 \\ 2 & 0 & 1\\ 4 & 0 & -1 \end{pmatrix}.\)
