Would this be a legal move in a Matrix? Here I have a matrix:
\begin{pmatrix}
    b^2c^2 & bc & b+c \\
    c^2a^2 & ca & c+a \\
    a^2b^2 & ab & a+b \\
    \end{pmatrix}
I was wondering can I do the following operation on this:
$C2 \to C2 \times C2$
So it would give me the following:
\begin{pmatrix}
    b^2c^2 & b^2c^2 & b+c \\
    c^2a^2 & c^2a^2 & c+a \\
    a^2b^2 & a^2b^2 & a+b \\
    \end{pmatrix}
This is needed in order to prove the determinant is equal to zero.

You need to think in terms of homogeneous symmetric polynomials' degrees. Subtract $ab+bc+ca$ times the second column from the first, thus changing $b^2c^2$ to $-abc(b+c)$. Now add $abc$ times the third column to the first; it should vanish then. Explicitly$$\left(\begin{array}{c}
b^{2}c^{2}\\
c^{2}a^{2}\\
a^{2}b^{2}
\end{array}\right)-\left(ab+bc+ca\right)\left(\begin{array}{c}
bc\\
ca\\
ab
\end{array}\right)+abc\left(\begin{array}{c}
b+c\\
c+a\\
a+b
\end{array}\right)=\left(\begin{array}{c}
0\\
0\\
0
\end{array}\right).$$
