Question

The determinants $\begin {vmatrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end {vmatrix}$ and $\begin {vmatrix} 1 & a & a^2 \\ 1 & b & b^2\\ 1 & c & c^2 \end {vmatrix} $ are not
identically equal .

Answer 1

Let $\Delta= \begin {vmatrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end {vmatrix} =\frac{1}{abc} \begin {vmatrix} a & a^2 & abc \\ b & b^2 & abc \\ c & c^2 & abc \end {vmatrix} $
Applying $R_1 \rightarrow \ aR_1,R_2 \ \rightarrow \ bR_2,R_3 \rightarrow \ cR_3$
$=\frac{1}{abc}.abc \begin {vmatrix} a & a^2 & 1 \\ b & b^2 & 1 \\ c & c^2 & 1 \end {vmatrix}= \begin {vmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \end {vmatrix}$
$\therefore \begin {vmatrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end {vmatrix} $ and $\begin {vmatrix} 1 & a & a^2 \\ 1 & b & b^2\\ 1 & c & c^2 \end {vmatrix} $
Hence , statement is false