1. **State the problem:** Find the adjoint (also called adjugate) of the matrices $$A = \begin{bmatrix} 3 & 5 \\ 3 & 4 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} 2 & 3 \\ 1 & -2 \end{bmatrix}$$ 2. **Recall the formula for the adjoint of a 2x2 matrix:** For a matrix $$M = \begin{bmatrix} a & b \\ c & d \end{bmatrix},$$ its adjoint is $$\text{adj}(M) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}.$$ This is the transpose of the cofactor matrix. 3. **Calculate the adjoint of matrix A:** Given $$A = \begin{bmatrix} 3 & 5 \\ 3 & 4 \end{bmatrix},$$ apply the formula: $$\text{adj}(A) = \begin{bmatrix} 4 & -5 \\ -3 & 3 \end{bmatrix}.$$ 4. **Calculate the adjoint of matrix B:** Given $$B = \begin{bmatrix} 2 & 3 \\ 1 & -2 \end{bmatrix},$$ apply the formula: $$\text{adj}(B) = \begin{bmatrix} -2 & -3 \\ -1 & 2 \end{bmatrix}.$$ 5. **Final answers:** $$\boxed{\text{adj}(A) = \begin{bmatrix} 4 & -5 \\ -3 & 3 \end{bmatrix}, \quad \text{adj}(B) = \begin{bmatrix} -2 & -3 \\ -1 & 2 \end{bmatrix}}$$