1. The statement is det(A - B) = det(A) - det(B). We need to determine if this is true or false. 2. Recall the property of determinants: $$\det(A + B) \neq \det(A) + \det(B)$$ in general, which means the determinant is not linear over matrix addition or subtraction. 3. Therefore, $$\det(A - B) \neq \det(A) - \det(B)$$ in general. 4. Answer: False. 5. Given a 3 \times 3 non-singular matrix A and using Cramer's rule to solve $$Ax = e_2$$, where $$e_2 = [0, 1, 0]^T$$ is the second standard basis vector. 6. Cramer's rule finds the $$j$$-th column of $$A^{-1}$$ by solving $$Ax = e_j$$. 7. Here, $$j = 2$$, so the solution $$x$$ is the second column of $$A^{-1}$$. 8. Answer: Second column. 9. The cross product of vectors $$\mathbf{u} = [-1, 2, -1]^T$$ and $$\mathbf{v} = [-2, -2, 0]^T$$ is given by: $$ \mathbf{u} \times \mathbf{v} = \begin{bmatrix} u_2 v_3 - u_3 v_2 \\ u_3 v_1 - u_1 v_3 \\ u_1 v_2 - u_2 v_1 \end{bmatrix} $$ 10. Compute each component: - First component: $$2 \times 0 - (-1) \times (-2) = 0 - 2 = -2$$ - Second component: $$-1 \times (-2) - (-1) \times 0 = 2 - 0 = 2$$ - Third component: $$-1 \times (-2) - 2 \times (-2) = 2 + 4 = 6$$ 11. The cross product vector is $$[-2, 2, 6]^T$$. 12. The second coordinate is $$2$$. 13. Answer: 2.