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(B) The system of equations is given by: $ax + by + cz = 2$ $bx + cy + az = 2$ $cx + ay + bz = 2$ The determinant of the coefficient matrix is: $\Delt

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is inconsistent

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(B) The system of equations is given by: $ax + by + cz = 2$ $bx + cy + az = 2$ $cx + ay + bz = 2$ The determinant of the coefficient matrix is: $\Delta = \begin{vmatrix} a & b & c \ b & c & a \ c & a & b \end{vmatrix}$ Applying the row operation $R_1 ightarrow R_1 + R_2 + R_3$: $\Delta = \begin{vmatrix} a+b+c & a+b+c & a+b+c \ b & c & a \ c & a & b \end{vmatrix}$ Since $a+b+c = 0$, we have: $\Delta = (a+b+c) \begin{vmatrix} 1 & 1 & 1 \ b & c & a \ c & a & b \end{vmatrix} = 0 	imes \begin{vmatrix} 1 & 1 & 1 \ b & c & a \ c & a & b \end{vmatrix} = 0$ Since $\Delta = 0$, the system either has no solution or infinitely many solutions. Let us check the consistency by calculating the determinants $\Delta_x, \Delta_y, \Delta_z$. $\Delta_x = \begin{vmatrix} 2 & b & c \ 2 & c & a \ 2 & a & b \end{vmatrix} = 2 \begin{vmatrix} 1 & b & c \ 1 & c & a \ 1 & a & b \end{vmatrix} = 2(c^2 + ab + ab - c^2 - a^2 - b^2) = 2(2ab - a^2 - b^2 - c^2)$. Since $a+b+c=0$, $c = -(a+b)$, so $c^2 = a^2 + b^2 + 2ab$. Thus, $\Delta_x = 2(2ab - a^2 - b^2 - (a^2 + b^2 + 2ab)) = 2(-2a^2 - 2b^2) = -4(a^2 + b^2)$. If $a, b, c$ are not all zero, $\Delta_x 
eq 0$. Since $\Delta = 0$ and at least one of $\Delta_x, \Delta_y, \Delta_z 
eq 0$, the system is inconsistent.

Consider the system of equations: $ax + by + cz = 2$, $bx + cy + az = 2$, $cx + ay + bz = 2$, where $a, b, c$ are real numbers such that $a + b + c = 0$. Then, the system

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