Row replacement operation not changing the determinant Can someone prove why a row replacement operation does not change the determinant of a matrix?
**row replacement operation being adding one row to another or something of that sort

If you have a matrix $\mathrm A \in \mathbb{R}^{n \times n}$ and you replace its $i$-th row by a linear combination of the $n$ rows of $\mathrm A$, this corresponds to left-multiplying $\mathrm A$ by
$$\mathrm E := \mathrm I_n - \mathrm{e}_i \mathrm{e}_i^T + \mathrm{e}_i \mathrm{c}^T = \mathrm I_n + \mathrm{e}_i (\mathrm{c} - \mathrm{e}_i)^T$$
where $\mathrm{e}_i$ is a vector with $n-1$ zeros and with a one on the $i$-th entry, and the entries of $\mathrm{c}$ are the coefficients of the linear combination. Thus, the determinant of the new matrix is
$$\det (\mathrm E \mathrm A) = \det(\mathrm E) \cdot \det (\mathrm A)$$
Using Weinstein-Aronszajn determinant identity,
$$\det (\mathrm E) = \det (\mathrm I_n + \mathrm{e}_i (\mathrm{c} - \mathrm{e}_i)^T) = \det (1 + (\mathrm{c} - \mathrm{e}_i)^T \mathrm{e}_i) = 1 + (c_i - 1) = c_i$$
Hence, replacing a row by a linear combination of rows does change the determinant if the coefficient that multiplies the row being replaced is not $1$. For example, replacing a row by the same row multiplied by $2$ then doubles the determinant.
