The ordered pair (a, b), for which the system of linear equations
$3 x - 2 y + z = b; 5 x - 8 y + 9 z = 3; 2 x + y + a z = - 1$
has no solution, is :

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$(- 3, - \frac{1}{3})$

$(3, \frac{1}{3})$

$(3, - \frac{1}{3})$

$(- 3, \frac{1}{3})$

answer is C.

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Detailed Solution

$\begin{array}{l} |\begin{array}{ccc} 3 & - 2 & 1 \\ 5 & - 8 & 9 \\ 2 & 1 & a \end{array}| = 0 \\ 3 (- 8 a - 9) + 2 (5 a - 18) + 1 (21) = 0 \\ \Rightarrow a = - 3 \end{array}$

$Δ_{2} = |\begin{array}{ccc} 3 & - 2 & b \\ 5 & 8 & 3 \\ 2 & 1 & - 1 \end{array}| Δ_{2} = 0 If b = \frac{1}{3}$

the system of linear equations has no solution $\Rightarrow a = - 3, b \neq \frac{1}{3}$

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