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Let $A = [\begin{matrix} x + y & y \\ 2 x & x - y \end{matrix}], B = [\begin{matrix} 2 \\ - 1 \end{matrix}] and C = [\begin{matrix} 3 \\ 2 \end{matrix}]$ If AB = C, then what is the value of the determinant of the matrix A ?

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32.
Let p, q and r be three distinct positive real numbers.
If D = $|\begin{matrix} p & q & r \\ q & r & p \\ r & p & q \end{matrix}|$ then which one of the following is correct ?

D < 0

D ≤ 0

D > 0

D ≥ 0

D < 0

$D = |\begin{matrix} p & q & r \\ q & r & p \\ r & p & q \end{matrix}| \Rightarrow D = p (qr - p^{2}) - q (q^{2} - pr) + r (qp - r^{2}) \Rightarrow D = pqr - p^{3} - q^{3} + pqr - r^{3} \Rightarrow D = 3 pqr - (p^{3} + q^{3} + r^{3}) Given that p, q and r be three distinct positive real numbers . Let p = 1, q = 2, r = 3 then 1^{3} + 2^{3} + 3^{3} = 36 and 3 \times 1 \times 2 \times 3 = 18 As p^{3} + q^{3} + r^{3} > 3 pqr So, D < 0$