What is the value of left|begin{array}{ccc}a | vedclass

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(B) Let $\Delta = \begin{vmatrix} a & b & c \ a-b & b-c & c-a \ b+c & c+a & a+b \end{vmatrix}$Applying $R_2 	o R_2 - R_1$:$\Delta = \begin{vmatrix}

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$a^3+b^3+c^3-3abc$

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(B) Let $\Delta = \begin{vmatrix} a & b & c \ a-b & b-c & c-a \ b+c & c+a & a+b \end{vmatrix}$Applying $R_2 	o R_2 - R_1$:$\Delta = \begin{vmatrix} a & b & c \ -b & -c & -a \ b+c & c+a & a+b \end{vmatrix}$Applying $R_3 	o R_3 + R_2$:$\Delta = \begin{vmatrix} a & b & c \ -b & -c & -a \ c & a & b \end{vmatrix}$Wait,let us re-evaluate the operation $R_3 	o R_3 + R_2$:$R_3$ becomes $(b+c-b, c+a-c, a+b-a) = (c, a, b)$.So,$\Delta = \begin{vmatrix} a & b & c \ -b & -c & -a \ c & a & b \end{vmatrix}$Expanding along $R_1$:$\Delta = a(-bc - (-a^2)) - b(-b^2 - (-ac)) + c(-ab - (-c^2))$$\Delta = a(a^2 - bc) - b(ac - b^2) + c(c^2 - ab)$$\Delta = a^3 - abc - abc + b^3 + c^3 - abc$$\Delta = a^3 + b^3 + c^3 - 3abc$

What is the value of $\left|\begin{array}{ccc}a & b & c \\ a-b & b-c & c-a \\ b+c & c+a & a+b\end{array}\right|=$ ?

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