I came across the following problem from the book Higher Algebra(by barnard and Child) that says:
Prove that $\,\,\begin{vmatrix} bc &bc'+b'c &b'c' \\ ca& ca'+c'a &c'a' \\ ab&ab'+a'b &a'b' \end{vmatrix}=(bc')(ca')(ab')$
My Try: Diving the first row by $b'c'$ ,the second by $c'a'$ ,the third by $a'b'$ and then putting $a/a'=x,b/b'=y,c/c'=z$ we get
$(a'b')(b'c')(c'a')\begin{vmatrix} yz & y+z &1 \\ zx& z+x & 1 \\ xy& x+y & 1 \end{vmatrix}=.....=(ab'-a'b)(bc'-b'c)(a'c-ac')$
That is the ultimate result I get. Can someone help?