Strange row reduction method

Question

This is a determinant problem with the solution. But I don't understand what sort of row reduction is being used or why. I can see the first term of each row is being subtracted from the others, but why?

Answer 1

The first step $$ {\det}\!\pmatrix{ 1&1&1\\ x&y&z\\ x^2&y^2&z^2\\ } = {\det}\!\pmatrix{ 1&0&0\\ x&y-x&z-x\\ x^2&y^2-x^2&z^2-x^2\\ } $$ is a column reduction, not a row reduction. The first column is subtracted from each of the other columns.

As to why it's a good move? Leaving only one nonzero entry in the first row instantly reduces the problem to a $2{\times}2$ determinant. Moreover, it produces common factors in the second and third columns, so it's not just a good move, it's a great move.

The next step $$ {\det}\!\pmatrix{ 1&0&0\\ x&y-x&z-x\\ x^2&y^2-x^2&z^2-x^2\\ } = (y-x)(z-x)\;{\det}\!\pmatrix{ 1&0&0\\ x&1&1\\ x^2&y+x&z+x\\ } $$ removes the common factor $y-x$ from the second column, and the common factor $z-x$ from the third column.

The next step $$ (y-x)(z-x)\;{\det}\!\pmatrix{ 1&0&0\\ x&1&1\\ x^2&y+x&z+x\\ } = (y-x)(z-x)\;{\det}\!\pmatrix{ 1&1\\ y+x&z+x\\ } $$ is expansion along the first row.

Expanding the $2{\times}2$ determinant yields the final answer.