Matrices and determinants question.

Question

Establish that if A is the matrix \begin{bmatrix} b+c & a^2 & a \\ c+a & b^2 & b \\ a+b & c^2 & c \\ \end{bmatrix} then $|A| = -(a-b)(b-c)(c-a)(a+b+c)$.

Answer 1

using linearity with respect to the first column, those matrix have the same determinant:

\begin{bmatrix} b+c & a^2 & a \\ c+a & b^2 & b \\ a+b & c^2 & c \\ \end{bmatrix} \begin{bmatrix} b+c + a& a^2 & a \\ c+a + b& b^2 & b \\ a+b +c & c^2 & c \\ \end{bmatrix} whose determinant is $a+b +c$ multiplied by \begin{bmatrix} 1 & a^2 & a \\ 1 & b^2 & b \\ 1 & c^2 & c \\ \end{bmatrix} which is $-1 \times$ the Vandermonde determinant.