To compute the determinant of the given matrix $A$ using cofactor expansion, we expand along the first row. Recall that the determinant of a $3 \times 3$ matrix is computed as:

$\text{det}(A) = a_{11} \text{det}(M_{11}) - a_{12} \text{det}(M_{12}) + a_{13} \text{det}(M_{13})$

Here, $M_{ij}$ is the minor matrix obtained by removing the $i$-th row and $j$-th column. The given matrix is:

-3a & -3 & -2 \\ -2 & 4b & -2 \\ 3 & 1 & -3c \end{bmatrix}$$ ### Step 1: Expand along the first row The cofactor expansion along the first row is: $$\text{det}(A) = (-3a) \text{det}(M_{11}) - (-3) \text{det}(M_{12}) + (-2) \text{det}(M_{13})$$ #### Compute $$\text{det}(M_{11})$$: The minor matrix $$M_{11}$$ is obtained by removing the first row and the first column: $$M_{11} = \begin{bmatrix} 4b & -2 \\ 1 & -3c \end{bmatrix}$$ The determinant of $$M_{11}$$ is: $$\text{det}(M_{11}) = (4b)(-3c) - (-2)(1) = -12bc + 2$$ #### Compute $$\text{det}(M_{12})$$: The minor matrix $$M_{12}$$ is obtained by removing the first row and the second column: $$M_{12} = \begin{bmatrix} -2 & -2 \\ 3 & -3c \end{bmatrix}$$ The determinant of $$M_{12}$$ is: $$\text{det}(M_{12}) = (-2)(-3c) - (-2)(3) = 6c + 6$$ #### Compute $$\text{det}(M_{13})$$: The minor matrix $$M_{13}$$ is obtained by removing the first row and the third column: $$M_{13} = \begin{bmatrix} -2 & 4b \\ 3 & 1 \end{bmatrix}$$ The determinant of $$M_{13}$$ is: $$\text{det}(M_{13}) = (-2)(1) - (4b)(3) = -2 - 12b$$ ### Step 2: Substitute into the expansion formula Substituting the computed values of $$\text{det}(M_{11})$$, $$\text{det}(M_{12})$$, and $$\text{det}(M_{13})$$ into the cofactor expansion formula: $$\text{det}(A) = (-3a)(-12bc + 2) - (-3)(6c + 6) + (-2)(-2 - 12b)$$ Simplify each term: 1. First term: $$(-3a)(-12bc + 2) = 36abc - 6a$$ 2. Second term: $$(-3)(6c + 6) = -18c - 18$$ 3. Third term: $$(-2)(-2 - 12b) = 4 + 24b$$ Combine all terms: $$\text{det}(A) = (36abc - 6a) - (18c + 18) + (4 + 24b)$$ $$\text{det}(A) = 36abc - 6a - 18c - 18 + 4 + 24b$$ $$\text{det}(A) = 36abc - 6a + 24b - 18c - 14$$ ### Final Answer: $$\text{det}(A) = 36abc - 6a + 24b - 18c - 14$$ Would you like a more detailed explanation of any step? Here are 5 related questions to consider: 1. What is the geometric significance of a matrix determinant? 2. How does cofactor expansion generalize to larger matrices? 3. What happens to the determinant if a matrix row is multiplied by a scalar? 4. How can properties of determinants simplify calculations for specific types of matrices? 5. What role do determinants play in solving systems of linear equations? **Tip:** When performing cofactor expansion, choose the row or column with the most zeros to simplify calculations!