C V Durell’s Advanced Algebra, which was a standard sixth-form textbook until at least 1965, explains the “row operations” as the very first item in its chapter on determinants.
It explains them first for 2×2 determinants, then later proves they also work for 3×3. They can be proved to work for n×n using the Levi-Civita symbol.
One reason why Durell gave more attention to “row operations” was that no-one had calculators then. If youi wanted to work out:
you either had to simplify the working-out with “row operations”, or do long multiplication.
But “row operations” are useful not just for calculation but also for understanding what’s going on, for example, with the scalar triple product
Or to see how the vector cross-product b×a = − a×b.
Or to identify determinants as calculating to zero when checking whether a particular number is an eigenvalue for a matrix.
Or, of course, to see how to calculate 4×4 and bigger determinants from “reduced row echelon form” (3×3 determinants are small enough to be calculated more conveniently by the “diagonals” method (as below).