1. **Problem statement:** We are given four statements about determinants of $n \times n$ matrices $A$ and $B$. We need to check which statements are true. 2. **Statement A:** $\det(A + B) = \det A + \det B$. - This is generally false. Determinant is not additive over matrix addition. - Example: For $A=I$ (identity), $B=-I$, $\det(A + B) = \det(0)=0$ but $\det A + \det B = 1 + 1 = 2$. 3. **Statement B:** The determinant of $A$ equals the product of the pivots in any echelon form $U$ of $A$, multiplied by $(-1)^r$, where $r$ is the number of row interchanges during row reduction from $A$ to $U$. - This is true. - Each row interchange changes determinant sign, so multiply by $(-1)^r$. - The pivots (diagonal elements) product equals determinant of the echelon matrix $U$. - Because determinant is multiplicative under elementary operations, this formula holds. 4. **Statement C:** Adding a multiple of one row to another does not affect the determinant. - True. - This row operation corresponds to an elementary matrix with determinant 1, so determinant remains unchanged. 5. **Statement D:** If columns of $A$ are linearly dependent, then $\det A = 0$. - True. - Linearly dependent columns imply $A$ is singular, so determinant is zero. **Final answer:** Statements B, C, and D are true while A is false.