Would this be a legal move in a Matrix?

Question

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.

Answer 1

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$$\left(\begin{array}{c} b^{2}c^{2}\\ c^{2}a^{2}\\ a^{2}b^{2} \end{array}\right)-\left(ab+bc+ca\right)\left(\begin{array}{c} bc\\ ca\\ ab \end{array}\right)+abc\left(\begin{array}{c} b+c\\ c+a\\ a+b \end{array}\right)=\left(\begin{array}{c} 0\\ 0\\ 0 \end{array}\right).$$

Answer 2

Since this will (for most $a$ and $b$) change the value of the determinant, lets use co-factor multiplication by your matrix $M$ to prove this, which shouldn't be too bad given it is a $3x3$ matrix. I will denote each $2x2$ co factor as $A_i$ for simplicity. $$det(M) = (b+c)det(A_1) - (c+a)det(A_2) + (a+b)det(A_3)$$ $$det(M) = (b+c)(a^3c^2b-a^3b^2c)-(c+a)(c^3b^2a-c^3a^2b)+(a+b)(b^3c^2a -b^3a^2c)$$ Hmm... maybe this wasn't the right approach but whatever too late now $$det(M) = (a^3c^2b^2-a^3b^3c+a^3c^3b-a^3b^2c^2)-(b^3c^3a-b^3a^2c^2+b^3c^2a^2-b^3a^3c)+(c^3b^2a^2-c^3a^3b+c^3b^3a-c^3a^2b^2)$$ Jesus christ... notice that we can take out a $a^3, b^3$ or $c^3$ $$det(M) = a^3(c^2b^2-b^3c+c^3b-b^2c^2)-b^3(c^3a-a^2c^2+c^2a^2-a^3c)+c^3(b^2a^2-a^3b+b^3a-a^2b^2)$$ And then do even better by noticing we can take out a $ab, bc$ or $ac$ $$det(M) = a^3bc(cb-b^2+c^2-bc)-b^3ca(c^2-ac+ca-a^2)+c^3ba(ba-a^2+b^2-ab)$$ Now some terms cancel... thank god.... $$det(M) = a^3bc(-b^2+c^2)-b^3ca(c^2-a^2)+c^3ba(-a^2+b^2)$$ Rearrange some more.. $$det(M) = -a^3bc(b^2-c^2)-b^3ca(c^2-a^2)-c^3ba(a^2-b^2)$$ Factor out a $-abc$ $$det(M) = -abc(a^2(b^2-c^2)+b^2(c^2-a^2)+c^2(a^2-b^2))$$ Distribute $$det(M) = -abc((a^2b^2-a^2c^2)+(b^2c^2-b^2a^2)+(c^2a^2-c^2b^2))$$ Reorder terms $$det(M) = -abc(a^2b^2-b^2a^2-a^2c^2+c^2a^2+b^2c^2-c^2b^2)$$ Thus $$det(M) = -abc(0)$$ $$det(M) = 0$$ I retract my former statement that this wouldn't be too bad, but luckily I denoted the co factors as $A_i$, saving me loads of time.