1. **State the problem:** We need to evaluate the determinant of the matrix $$\begin{pmatrix}3 & -2 & 2 \\ 1 & 2 & -3 \\ 4 & 1 & 2\end{pmatrix}$$ using Laplace expansion along the first row and then along the second row. 2. **Recall the formula for Laplace expansion:** The determinant of a matrix can be expanded along any row or column as $$\det(A) = \sum_{j=1}^n (-1)^{i+j} a_{ij} M_{ij}$$ where $a_{ij}$ is the element in the $i$-th row and $j$-th column, and $M_{ij}$ is the determinant of the submatrix obtained by deleting the $i$-th row and $j$-th column. 3. **Laplace expansion along the first row ($i=1$):** - For element $a_{11} = 3$, minor matrix is $$\begin{pmatrix}2 & -3 \\ 1 & 2\end{pmatrix}$$ with determinant $$2 \times 2 - (-3) \times 1 = 4 + 3 = 7$$ - For element $a_{12} = -2$, minor matrix is $$\begin{pmatrix}1 & -3 \\ 4 & 2\end{pmatrix}$$ with determinant $$1 \times 2 - (-3) \times 4 = 2 + 12 = 14$$ - For element $a_{13} = 2$, minor matrix is $$\begin{pmatrix}1 & 2 \\ 4 & 1\end{pmatrix}$$ with determinant $$1 \times 1 - 2 \times 4 = 1 - 8 = -7$$ Calculate the determinant: $$\det = 3 \times 7 - (-2) \times 14 + 2 \times (-7) = 21 + 28 - 14 = 35$$ 4. **Laplace expansion along the second row ($i=2$):** - For element $a_{21} = 1$, minor matrix is $$\begin{pmatrix}-2 & 2 \\ 1 & 2\end{pmatrix}$$ with determinant $$-2 \times 2 - 2 \times 1 = -4 - 2 = -6$$ - For element $a_{22} = 2$, minor matrix is $$\begin{pmatrix}3 & 2 \\ 4 & 2\end{pmatrix}$$ with determinant $$3 \times 2 - 2 \times 4 = 6 - 8 = -2$$ - For element $a_{23} = -3$, minor matrix is $$\begin{pmatrix}3 & -2 \\ 4 & 1\end{pmatrix}$$ with determinant $$3 \times 1 - (-2) \times 4 = 3 + 8 = 11$$ Calculate the determinant: $$\det = (-1)^{2+1} \times 1 \times (-6) + (-1)^{2+2} \times 2 \times (-2) + (-1)^{2+3} \times (-3) \times 11$$ $$= -1 \times 1 \times (-6) + 1 \times 2 \times (-2) - 1 \times (-3) \times 11 = 6 - 4 + 33 = 35$$ 5. **Final answer:** The determinant of the matrix is $$\boxed{35}$$ whether expanded along the first or second row.