1. **Problem Statement:** Find the determinant of the matrix $$A = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$$. 2. **Recall the determinant formula for a 2x2 matrix:** For $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix},$$ the determinant is $$\det(A) = ad - bc$$. 3. **Identify the elements:** Here, $$a = 1$$, $$b = 1$$, $$c = 1$$, and $$d = -1$$. 4. **Calculate:** $$\det(A) = (1)(-1) - (1)(1) = -1 - 1 = -2$$. **Final answer:** The determinant of matrix $$A$$ is $$-2$$.