Effect of row operations on the determinant

1. Swapping. The determinant changes sign: $\det = -\det_{\text{original}}$. This follows from the alternating property (the determinant is zero when two rows are equal, and swapping is an odd permutation).

2. Scaling. Multiplying a row by $\lambda$ multiplies the determinant by $\lambda$: $\det = \lambda\cdot\det_{\text{original}}$. This follows from linearity in each row.

3. Adding a multiple. Adding a multiple of one row to another leaves the determinant unchanged. Indeed, writing the new row as $L_i+\lambda L_j$, linearity gives $\det(L_i+\lambda L_j,\ldots)=\det(L_i,\ldots)+\lambda\det(L_j,\ldots)$, and the second determinant has two equal rows, so it is zero.

Conclusion. Swapping changes sign; scaling by $\lambda$ scales by $\lambda$; adding a multiple leaves the determinant unchanged. These rules are used to simplify determinant computations.