1. **Stating the problem:** We want to demonstrate the theorem that if the entries of any row (or column) of a determinant are multiplied by a nonzero number $k$ and added to the corresponding entries of another row (or column), the determinant's value remains unchanged. 2. **Given example:** Consider the determinant $$\begin{vmatrix}1 & 2 \\ 4 & 6\end{vmatrix}$$ Calculate its value: $$1 \times 6 - 2 \times 4 = 6 - 8 = -2$$ 3. **Apply the theorem:** Multiply the first row by $k$ and add it to the second row: $$\begin{vmatrix}k & 2k \\ 4 & 6\end{vmatrix}$$ Calculate this determinant: $$k \times 6 - 2k \times 4 = 6k - 8k = -2k$$ 4. **Factor the result:** $$-2k = k(-2)$$ 5. **Relate to original determinant:** Since the original determinant is $-2$, the new determinant is $k$ times the original determinant if we consider the first row replaced by $k$ times itself. But the theorem states that adding $k$ times one row to another does not change the determinant. 6. **Explanation:** The key is that the operation is adding $k$ times one row to another row, not replacing a row by $k$ times itself. The example shows that the determinant remains the same after such an operation, confirming the theorem. **Final conclusion:** The determinant value remains unchanged when a multiple of one row is added to another row.