I can't deal with a certain type of determinants problems

Question

Prove that, without expanding

How should my thinking start about these problems? how can I prove that the product of two variables equals square of the third?

Answer 1

Note that $$ \begin{vmatrix} 1&1&1\\ a&b&c\\ bc&ac&ba \end{vmatrix} = \frac 1{abc} \begin{vmatrix} 1&1&1\\ a&b&c\\ 1/a&1/b&1/c \end{vmatrix} = \frac 1{bc} \begin{vmatrix} a&1&1\\ a^2&b&c\\ 1&1/b&1/c \end{vmatrix} =\\ \frac 1{c} \begin{vmatrix} a&b&1\\ a^2&b^2&c\\ 1&1&1/c \end{vmatrix} = \begin{vmatrix} a&b&c\\ a^2&b^2&c^2\\ 1&1&1 \end{vmatrix} =\\ \begin{vmatrix} 1&1&1\\ a&b&c\\ a^2&b^2&c^2\\ \end{vmatrix} $$ and from here, it's enough to note that $|A| = |A^T|$. If we want to phrase this in terms of matrix multiplication, this amounts to the observation that $$ \pmatrix{0&0&1\\1&0&0\\0&1&0} \pmatrix{1\\&1\\&&\frac{1}{abc}}\pmatrix{1&1&1\\a&b&c\\bc&ac&ab} \pmatrix{a\\&b\\&&c} = \pmatrix{1&1&1\\ a&b&c\\ a^2&b^2&c^2} $$

Try to apply a similar trick to the second question. For the first three steps, note that $$ \begin{vmatrix} bc&a^2&a^2\\ b^2&ac&b^2\\ c^2&c^2&ab \end{vmatrix} = \frac{1}{abc} \begin{vmatrix} bc/a & a & a\\ b & ac/b & b\\ c & c & ab/c \end{vmatrix} $$