Can anyone teach me how to prove the identity without evaluating the determinant?

Question

I have searched through google yet i cant find a decent explaination. 

|a1 b1+ta1 c1+rb1+sa1| 

|a2 b2+ta2 c2+rb2+sa2| 

|a3 b3+ta3 c3+rb3+sa3| 

= 

|a1 a2 a3| 

|b1 b2 b3| 

|c1 c2 c3|

Answer 1

The value of the determinant is not changed by adding a scalar multiple of any column to another column.

For this reason the value of the whole determinant is the same as if the amounts added to each column were not added.

Then you need to know that the determinant of the transpose is equal to the original determinant.