1.
Let abcαβγaαbβcγ=0\begin{vmatrix} a & b & c \\ \alpha & \beta & \gamma \\ a-\alpha & b-\beta & c-\gamma \end{vmatrix} = 0, where aαa \neq \alpha, βb\beta \neq b, cγc \neq \gamma and a,b,c,α,β,γa, b, c, \alpha, \beta, \gamma are non-zero distinct real numbers. Then abγαbγaαb+γγc\begin{vmatrix} a & b & \gamma \\ \alpha & b & \gamma \\ a-\alpha & b+\gamma & \gamma-c \end{vmatrix} is equal to: