I have the following determinant :
Row1: (-2a) (a+b) (a+c)
Row2: (a+b) (-2b) (b+c)
Row3: (a+c) (b+c) (-2c)
I need to prove that the determinant equals 4(a+b)(b+c)(c+a).
I have tried using all the standard determinant properties, but I wasn't able to arrive at the final expression.
The result can be verified by expanding the determinant, but that is obviously too long. Another method is to realize that the determinant must be a homogeneous expression in a,b and c of third degree. Moreover, on replacing (a with -b) the determinant evaluates to zero, indicating that (a+b) must be a factor. Using this approach I concluded that the final answer must be of the form:
K(a+b)(b+c)(c+a).
The value of k can be determined by substituting some easy numbers for a,b and c such as 0,1 and 2.
But I am still not satisfied as I want to solve the determinant using its basic properties (such as row operations and factoring out terms). Please explain how that can be done. It should be possible right ?
Applying the Rule of Sarrus we obtain $$ \det (A)=4(a^2b + a^2c + ab^2 + 2abc + ac^2 + b^2c + bc^2)=4(a+b)(b+c)(c+a). $$ I don't think that this computation is "obviously too long". It is rather nice.