We start easy with a straightforward theorem whose proof presages the style of subsequent proofs in this subsection.
Suppose that $A$ is a square matrix with a row where every entry is zero, or a column where every entry is zero. Then $\detname{A}=0$.
Suppose that $A$ is a square matrix. Let $B$ be the square matrix obtained from $A$ by interchanging the location of two rows, or interchanging the location of two columns. Then $\detname{B}=-\detname{A}$.
So Theorem DRCS tells us the effect of the first row operation (Definition RO) on the determinant of a matrix. Here is the effect of the second row operation.
Suppose that $A$ is a square matrix. Let $B$ be the square matrix obtained from $A$ by multiplying a single row by the scalar $\alpha$, or by multiplying a single column by the scalar $\alpha$. Then $\detname{B}=\alpha\detname{A}$.
Let us go for understanding the effect of all three row operations. But first we need an intermediate result, but it is an easy one.
Suppose that $A$ is a square matrix with two equal rows, or two equal columns. Then $\detname{A}=0$.
Now explain the third row operation. Here we go.
Suppose that $A$ is a square matrix. Let $B$ be the square matrix obtained from $A$ by multiplying a row by the scalar $\alpha$ and then adding it to another row, or by multiplying a column by the scalar $\alpha$ and then adding it to another column. Then $\detname{B}=\detname{A}$.
Is this what you expected? We could argue that the third row operation is the most popular, and yet it has no effect whatsoever on the determinant of a matrix! We can exploit this, along with our understanding of the other two row operations, to provide another approach to computing a determinant. We'll explain this in the context of an example.