Let a, b, c notin {0, 1}. If the system of eq | vedclass

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(B) The given homogeneous system of equations is $AX=0$, where $A = \begin{bmatrix} 1 & a & a \ b & 1 & b \ c & c & 1 \end{bmatrix}$.Since the syste

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infinite number of solutions

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(B) The given homogeneous system of equations is $AX=0$, where $A = \begin{bmatrix} 1 & a & a \ b & 1 & b \ c & c & 1 \end{bmatrix}$.Since the system has a non-trivial solution, the determinant $|A| = 0$.Calculating the determinant: $1(1-bc) - a(b-bc) + a(bc-c) = 1 - bc - ab + abc + abc - ac = 1 - ab - bc - ca + 2abc = 0$.Now consider the non-homogeneous system $\Pi_1=a, \Pi_2=b, \Pi_3=c$. The augmented matrix is $A' = \begin{bmatrix} 1 & a & a & | & a \ b & 1 & b & | & b \ c & c & 1 & | & c \end{bmatrix}$.Since $|A|=0$, the system either has no solution or infinitely many solutions.By performing row operations, we observe that the rank of the augmented matrix $A'$ is equal to the rank of $A$ (which is $2$ for $a, b, c 
eq 1$).Since the rank of the coefficient matrix equals the rank of the augmented matrix, the system has infinitely many solutions.

Let $a, b, c \notin \{0, 1\}$. If the system of equations $\Pi_1 \equiv x+ay+az=0, \Pi_2 \equiv bx+y+bz=0, \Pi_3 \equiv cx+cy+z=0$ has a non-trivial solution, then the system of equations $\Pi_1=a, \Pi_2=b, \Pi_3=c$ has

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