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

One way to think about it, using the property: $\det(AB)=\det(A)\det(B)$:
Adding a multiple of one row to another is equivalent to left multiplication by an elementary matrix.
Let $B$ be some $n\times n$ matrix, $A$ be an $n\times n$ elementary matrix which acts as an operator which adds $k$ copies of row $i$ to row $j$. So applying that same row operation to $B$ will result in the matrix $AB$. Then without a loss of generality, $A$ has the form:
$$\begin{bmatrix}1 & & & & & \\ & \ddots & & & & & \\ & & 1 & & & & \\ & & & \ddots & & & \\ & & k & & 1 & & \\ & & & & & \ddots & \\ & & & & & & 1\end{bmatrix}$$
where $a_{ji}=k$.
The determinant of a triangular matrix is the product of the diagonal. $A$ has a unit diagonal, so $\det(A)=1$.
Therefore, $$\det(AB)=\det(A)\det(B)=1\det(B)=\det(B).$$
