1. **State the problem:** Determine if the matrix $$\begin{pmatrix}-1 & 0.5 \\ 4 & -2 \\ 4 & 1\end{pmatrix}$$ has full rank. 2. **Recall the definition:** The rank of a matrix is the maximum number of linearly independent rows or columns. For a matrix with dimensions $$3 \times 2$$, the maximum rank is $$2$$ (the smaller dimension). 3. **Check linear independence of columns:** The columns are $$\begin{pmatrix}-1 \\ 4 \\ 4\end{pmatrix}$$ and $$\begin{pmatrix}0.5 \\ -2 \\ 1\end{pmatrix}$$. 4. **Test if one column is a scalar multiple of the other:** Suppose $$\begin{pmatrix}0.5 \\ -2 \\ 1\end{pmatrix} = k \begin{pmatrix}-1 \\ 4 \\ 4\end{pmatrix}$$ for some scalar $$k$$. 5. Equate components: $$0.5 = -1 \times k \Rightarrow k = -0.5$$ $$-2 = 4 \times k = 4 \times (-0.5) = -2$$ (matches) $$1 = 4 \times k = 4 \times (-0.5) = -2$$ (does not match) 6. Since the third component does not satisfy the scalar multiple condition, the columns are linearly independent. 7. **Conclusion:** The matrix has rank $$2$$, which is full rank for a $$3 \times 2$$ matrix. **Final answer:** The matrix has full rank (rank = 2).