Q.
Let Δ=∣∣​AxByCz​x2y2z2​111​∣∣​ and Δ1​=∣∣​Axzy​Byzx​Czxy​∣∣​ then ∣∣​Axx21​Byy21​Cyz21​∣∣​
We have, Δ=⎣⎡​AxByCz​x2y2z2​111​⎦⎤​ and Δ1​=⎣⎡​Axzy​Byzx​Czxy​⎦⎤​
Now, Δ1​=⎣⎡​AXzy​Byzx​CZxy​⎦⎤​
On applying C1​→xC1​,C2​→yC2​,C3​→zC3​, we get =xyz1​⎣⎡​Axx2xyz​Byy2xyz​Czz2xyz​⎦⎤​
Taking common xyz from R3​ =xyzxyz​⎣⎡​Axx21​Byy21​Czz21​⎦⎤​ =⎣⎡​Axx21​Byy21​Czz21​⎦⎤​ =⎣⎡​AxByCz​x2y2z2​111​⎦⎤​ ∣∵∣A′∣A∣] =Δ