What will be value of \(\begin{vmatrix}2bc – a^2 & c^2 & b^2 \\c^2 & 2ac – b^2 & a^2 \\b^2 & a^2 & 2ab – c^2 \end {vmatrix}\) if given anor determinant \(\begin{vmatrix}a & b & c \\b & c & a \\c & a & b \end {vmatrix}\)?

Now, \(\begin{vmatrix}a & b & c \\b & c & a \\c & a & b \end {vmatrix}\)
Interchanging 2^nd and 3^rd columns,
= –\(\begin{vmatrix}a & c & b \\b & a & c \\c & b & a \end {vmatrix}\)
= \(\begin{vmatrix}-a & c & b \\ -b & a & c \\-c & b & a \end {vmatrix}\)
So, \(\begin{vmatrix}a & b & c \\b & c & a \\c & a & b \end {vmatrix}^2\) = \(\begin{vmatrix}a & b & c \\b & c & a \\c & a & b \end {vmatrix}\)\(\begin{vmatrix} -a & c & b \\ -b & a & c \\-c & b & a \end {vmatrix}\)
= {?(a³ + b³ + c³ ? 3abc)}² = \(\begin{vmatrix}-a^2 + bc + bc & -ab + ab + c^2 & -ac + b^2 + ca \\-ab + ab + c^2 & -ac – b^2 + ca & -a^2 + bc + bc \\-ac + b^2 + ca & -a^2 + bc + bc & -ab + ab – c^2 \end {vmatrix}\)
=> \(\begin{vmatrix}2bc – a^2 & c^2 & b^2 \\c^2 & 2ac – b^2 & a^2 \\b^2 & a^2 & 2ab – c^2 \end {vmatrix}\) = (a³ + b³ + c³ ? 3abc)²