To evaluate the given determinant Question: Evaluate the determinant 
$\left|
\begin{array}{cc} b^2c^2 & bc & b+c \\ 
c^2a^2 & ca & c+a \\ 
a^2b^2 & ab & a+b \\ 
\end{array}
\right|$
My answer:
$\left|
\begin{array}{cc} b^2c^2 & bc & b+c \\ 
c^2a^2 & ca & c+a \\ 
a^2b^2 & ab & a+b \\ 
\end{array}
\right|= \left|
\begin{array}{cc} b^2c^2 & bc & c \\ 
c^2a^2 & ca & a \\ 
a^2b^2 & ab & b \\ 
\end{array}
\right| + \left|
\begin{array}{cc} b^2c^2 & bc & b \\ 
c^2a^2 & ca & c \\ 
a^2b^2 & ab & a \\ 
\end{array}
\right|= abc \left|
 \begin{array}{cc} bc^2 & c & 1 \\ 
ca^2 & a & 1 \\ 
ab^2 & b & 1 \\ 
\end{array}
\right| +abc \left|
\begin{array}{cc} b^2c & b & 1 \\ 
c^2a & c & 1 \\ 
a^2b & a & 1 \\ 
\end{array}
\right|$
how do I proceed from here?

Your given matrix is : $\left|
\begin{array}{cc} b^2c^2 & bc & b+c \\ 
c^2a^2 & ca & c+a \\ 
a^2b^2 & ab & a+b \\ 
\end{array}
\right|$
Determinant of the given matrix is :
$\implies b^2c^2[ca^2 + abc - abc + a^2b] - bc[a^3c^2 + a^2bc^2 - a^2b^2c - a^3b^2] + (b+c)[a^3bc^2 - a^3b^2c]$
$\implies b^2c^2[ca^2  + a^2b] - bc[a^3c^2 + a^2bc^2 - a^2b^2c - a^3b^2] + (b+c)[a^3bc^2 - a^3b^2c]$
$\implies a^2b^2c^3 - a^2b^3c^2 - a^3bc^3 - a^2b^2c^3 + a^2b^3c^2 + a^3b^3c + a^3b^2c^2 - a^3b^3c + a^3bc^3 - a^3b^2c^2$
$\implies a^2b^2c^3 - a^2b^2c^3 - a^2b^3c^2 + a^2b^3c^2 - a^3bc^3 + a^3bc^3  + a^3b^3c - a^3b^3c + a^3b^2c^2  - a^3b^2c^2$
$\implies 0$
Therefore, given matrix is singular.
