1. State the problem: Find the determinant of the matrix $$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$. 2. Formula: For a 2x2 matrix $$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$$, the determinant is $$ad - bc$$. 3. Calculation: $$\det(A) = (1)(4) - (2)(3) = 4 - 6 = -2$$. 4. Explanation: Multiply the elements of the main diagonal and subtract the product of the other diagonal. 1. State the problem: Find the determinant of the matrix $$B = \begin{bmatrix} 2 & 0 & 1 \\ 3 & 0 & 0 \\ 5 & 1 & 1 \end{bmatrix}$$. 2. Formula: For a 3x3 matrix, use cofactor expansion along the first row: $$\det(B) = b_{11}C_{11} + b_{12}C_{12} + b_{13}C_{13}$$ where $$C_{ij}$$ are cofactors. 3. Calculation: $$\det(B) = 2 \times \det\begin{bmatrix} 0 & 0 \\ 1 & 1 \end{bmatrix} - 0 + 1 \times \det\begin{bmatrix} 3 & 0 \\ 5 & 1 \end{bmatrix}$$ $$= 2(0 \times 1 - 0 \times 1) + 1(3 \times 1 - 0 \times 5) = 0 + 3 = 3$$. 4. Explanation: Expand along the first row, calculate 2x2 determinants, then sum with signs. 1. State the problem: Find the determinant of $$C = \begin{bmatrix} 4 & 3 \\ 6 & 3 \end{bmatrix}$$. 2. Formula: $$\det(C) = ad - bc$$. 3. Calculation: $$\det(C) = (4)(3) - (3)(6) = 12 - 18 = -6$$. 4. Explanation: Same as previous 2x2 determinant calculation. 1. State the problem: Find the determinant of $$D = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{bmatrix}$$. 2. Formula: Use cofactor expansion along the first row. 3. Calculation: $$\det(D) = 1 \times \det\begin{bmatrix} 1 & 4 \\ 6 & 0 \end{bmatrix} - 2 \times \det\begin{bmatrix} 0 & 4 \\ 5 & 0 \end{bmatrix} + 3 \times \det\begin{bmatrix} 0 & 1 \\ 5 & 6 \end{bmatrix}$$ $$= 1(1 \times 0 - 4 \times 6) - 2(0 \times 0 - 4 \times 5) + 3(0 \times 6 - 1 \times 5)$$ $$= 1(0 - 24) - 2(0 - 20) + 3(0 - 5) = -24 + 40 - 15 = 1$$. 4. Explanation: Calculate each minor determinant carefully and apply signs. 1. State the problem: Find the determinant of $$E = \begin{bmatrix} 2 & -1 & 0 \\ 1 & 3 & 4 \\ 0 & 5 & 1 \end{bmatrix}$$. 2. Formula: Use cofactor expansion along the first row. 3. Calculation: $$\det(E) = 2 \times \det\begin{bmatrix} 3 & 4 \\ 5 & 1 \end{bmatrix} - (-1) \times \det\begin{bmatrix} 1 & 4 \\ 0 & 1 \end{bmatrix} + 0$$ $$= 2(3 \times 1 - 4 \times 5) + 1(1 \times 1 - 4 \times 0) + 0$$ $$= 2(3 - 20) + 1(1 - 0) = 2(-17) + 1 = -34 + 1 = -33$$. 4. Explanation: Expand along the first row, calculate minors, and sum with correct signs.