Determinant proof using its properties

Question

Prove without expanding: \begin{equation} \begin{vmatrix}bc&a^2&a^2\\b^2&ac&b^2\\c^2&c^2 & ab\end{vmatrix} = \begin{vmatrix}ac&bc&ab\\bc&ab&ac\\ab&ac&bc\end{vmatrix} \end{equation}

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Tried to multiply by 'abc' in all rows then take common factors.

Tried to expand determinant into two determinants.

I used the determinant properties shown here: http://www.vitutor.com/alg/determinants/properties_determinants.html

Answer 1

We check that the property holds when one of $a,b,c$ is zero. Now assume $a,b,c$ are all non-zero.

We divide the columns by $bc$, $ac$, and $ab$, respectively, and multiply the rows by $bc$, $ac$, and $ab$ respectively. This gives

$$\begin{vmatrix}bc&a^2&a^2\\b^2&ac&b^2\\c^2&c^2 & ab\end{vmatrix} = \begin{vmatrix}bc&ab&ac\\ab&ac&bc\\ac&bc&ab\end{vmatrix}$$ then we cycle the rows to obtain $$=\begin{vmatrix}ac&bc&ab\\bc&ab&ac\\ab&ac&bc\end{vmatrix}.$$