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(A) Given the determinant $\Delta = \left| \begin{array}{ccc} a & b & a - b \ b & c & b - c \ 2 & 1 & 0 \end{array} ight| = 0$. Expanding along th

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$G. P.$

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(A) Given the determinant $\Delta = \left| \begin{array}{ccc} a & b & a - b \ b & c & b - c \ 2 & 1 & 0 \end{array} ight| = 0$. Expanding along the third row: $2 \cdot \left| \begin{array}{cc} b & a - b \ c & b - c \end{array} ight| - 1 \cdot \left| \begin{array}{cc} a & a - b \ b & b - c \end{array} ight| + 0 = 0$. $2(b(b - c) - c(a - b)) - (a(b - c) - b(a - b)) = 0$. $2(b^2 - bc - ac + bc) - (ab - ac - ab + b^2) = 0$. $2(b^2 - ac) - (b^2 - ac) = 0$. $b^2 - ac = 0$. $b^2 = ac$. This condition implies that $a, b, c$ are in $G. P.$

The determinant $\left| \begin{array}{ccc} a & b & a - b \\ b & c & b - c \\ 2 & 1 & 0 \end{array} \right|$ is equal to zero if $a, b, c$ are in

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