Elementary row operations are really important when we want to find the determinant of a matrix. They help to make the problem easier without changing the basic properties we need. These operations include:

1. Row Swapping: This means switching two rows of a matrix. When you do this, the determinant changes its sign. So, if you swap rows an odd number of times, the new determinant will be the opposite of the original one. This can be helpful when we need to rearrange rows to make calculations simpler.

2. Scaling a Row: If you take a row and multiply it by a non-zero number (let's call it $k$), the determinant of the whole matrix is also multiplied by that same number $k$. So, we need to keep note of this change to find the right determinant.

3. Row Addition: This operation is when you add a multiple of one row to another row. Doing this does not change the determinant at all. This is really helpful for making zeros in a column, which makes it easier to work on calculations, especially when trying to get the matrix into a triangular shape.

Using these operations, we can change a matrix into an upper triangular form. This shape makes it much easier to find the determinant because we can simply multiply the numbers along the diagonal. If we have an $n \times n$ matrix in this triangular form, the determinant is found by this simple formula:

$$\text{det}(A) = a_{11} a_{22} \cdots a_{nn}$$

Here, $a_{ii}$ are the numbers on the diagonal of the triangular matrix.

In short, elementary row operations are key tools for finding the determinant. They help us simplify matrices while keeping everything we need for correct calculations. Getting good at these operations makes calculating determinants much faster and easier in linear algebra.