\[\left|\begin{array}{ccc}b+c & a+b & a \\c+a & b+c & b \\a+b & c+a &

1.	\[\left\|\begin{array}{ccc}b+c & a+b & a \\c+a & b+c & b \\a+b & c+a & c\end{array}\right\|\]1) \( a^{3}+b^{3}+c^{3}-3 a b c \)2) \( 3 a b c-a^{3}-b^{3}-c^{3} \)3) \( 2\left(a^{3}+b^{5}+c^{3}-3 a b c\right) \)4) \( 2\left(3 a b c-a^{3}-b^{3}-c^{3}\right. \)
Answer» \(\left\|\begin{array}{ccc} b+c & a+b & a \\ c+a & b+c & b \\ a+b & c+a & c \end{array}\right\|\) \(= \) \((b + c) \begin{bmatrix}b + c&b\\c + a&c\end{bmatrix} - (a + b)\begin{bmatrix}c + a&b\\a + b&c\end{bmatrix} + a\begin{bmatrix}c + a&b + c\\a + b&a + c\end{bmatrix}\) \( = (b + c)[(bc + c^2) - (bc + ab)] - [(a + b)(c^2 - b^2 + ac - ab)] + a[(a^2 + 2ac + c^2 - (ab + ac + b^2 + bc)]\) \(= (b + c)(c^2 - ab) - [(a + b)(c^2 - b^2 + ac - ab)] + a(a^2 - b^2 + c^2 + ac - ab - bc)\) \( = bc^2 - bc^2 - ab^2 - ab^2 + ab^2 + ab^2 - ac^2 + ac^2 - a^2c + a^2c + a^2b - a^2b + a^3 + b^3 + c^3 - 3abc\) \( = a^3 + b^3 + c^3 - 3abc\) Option (1) is correct.

\[\left|\begin{array}{ccc}b+c & a+b & a \\c+a & b+c & b \\a+b & c+a & c\end{array}\right|\]1) \( a^{3}+b^{3}+c^{3}-3 a b c \)2) \( 3 a b c-a^{3}-b^{3}-c^{3} \)3) \( 2\left(a^{3}+b^{5}+c^{3}-3 a b c\right) \)4) \( 2\left(3 a b c-a^{3}-b^{3}-c^{3}\right. \)

Answer»

\(\left|\begin{array}{ccc} b+c & a+b & a \\ c+a & b+c & b \\ a+b & c+a & c \end{array}\right|\)

\(= \) \((b + c) \begin{bmatrix}b + c&b\\c + a&c\end{bmatrix} - (a + b)\begin{bmatrix}c + a&b\\a + b&c\end{bmatrix} + a\begin{bmatrix}c + a&b + c\\a + b&a + c\end{bmatrix}\)

\( = (b + c)[(bc + c^2) - (bc + ab)] - [(a + b)(c^2 - b^2 + ac - ab)] + a[(a^2 + 2ac + c^2 - (ab + ac + b^2 + bc)]\)

\(= (b + c)(c^2 - ab) - [(a + b)(c^2 - b^2 + ac - ab)] + a(a^2 - b^2 + c^2 + ac - ab - bc)\)

\( = bc^2 - bc^2 - ab^2 - ab^2 + ab^2 + ab^2 - ac^2 + ac^2 - a^2c + a^2c + a^2b - a^2b + a^3 + b^3 + c^3 - 3abc\)

\( = a^3 + b^3 + c^3 - 3abc\)

Option (1) is correct.

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