If a1- a2- a3- b1- b2- b3 and aibj- 1 for 1- i- j- 3- then the determinant - -1-a13b13-1-a1b1 1-a13b23-1-a1b2 1-a13b33-1-a1b3 1-a23b13-1-a2b1 1-a23b23-1-a2b2 1-a23b33-1-a2b3 1-a33b13-1-a1b1 1-a33b23-1-a3b2 1-a33b33-1-a3b3is

The correct option is A positive 1 a_1^3b_1^3/1 a_1b_1 amp; 1 a_1^3b_2^3/1 a_1b_2 amp; 1 a_1^3b_3^3/1 a_1b_3 nbsp;1 a_2^3b_1^3/1 a_2b_1 amp; 1 ...

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If a1> a2> ...

Question

If $a_{1} > a_{2} > a_{3}$ , $b_{1} > b_{2} > b_{3}$ and $a_{i} b_{j} \neq 1$ for $1 \leq i$ , $j \leq 3$ , then the determinant
$Δ = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \frac{1 - a_{1}^{3} b_{1}^{3}}{1 - a_{1} b_{1}} & \frac{1 - a_{1}^{3} b_{2}^{3}}{1 - a_{1} b_{2}} & \frac{1 - a_{1}^{3} b_{3}^{3}}{1 - a_{1} b_{3}} \frac{1 - a_{2}^{3} b_{1}^{3}}{1 - a_{2} b_{1}} & \frac{1 - a_{2}^{3} b_{2}^{3}}{1 - a_{2} b_{2}} & \frac{1 - a_{2}^{3} b_{3}^{3}}{1 - a_{2} b_{3}} \frac{1 - a_{3}^{3} b_{1}^{3}}{1 - a_{1} b_{1}} & \frac{1 - a_{3}^{3} b_{2}^{3}}{1 - a_{3} b_{2}} & \frac{1 - a_{3}^{3} b_{3}^{3}}{1 - a_{3} b_{3}} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣$
is

positive

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non-negative

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negative

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non-positive

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Similar questions

a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} \in R

a_{i} b_{j} \neq 1

1 \leq i, j \leq 3

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \frac{1 - a_{1}^{3} b_{1}^{3}}{1 - a_{1} b_{1}} & \frac{1 - a_{1}^{3} b_{2}^{3}}{1 - a_{1} b_{2}} & \frac{1 - a_{1}^{3} b_{3}^{3}}{1 - a_{1} b_{3}} \frac{1 - a_{2}^{3} b_{1}^{3}}{1 - a_{2} b_{1}} & \frac{1 - a_{2}^{3} b_{2}^{3}}{1 - a_{2} b_{2}} & \frac{1 - a_{2}^{3} b_{3}^{3}}{1 - a_{2} b_{3}} \frac{1 - a_{3}^{3} b_{1}^{3}}{1 - a_{3} b_{1}} & \frac{1 - a_{3}^{3} b_{2}^{3}}{1 - a_{3} b_{2}} & \frac{1 - a_{3}^{3} b_{3}^{3}}{1 - a_{3} b_{3}} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

a_{1} < a_{2} < a_{3}

b_{1} < b_{2} < b_{3}

a_{1} > a_{2} a_{3}

b_{1} > b_{2} > b_{3}

(a_{1} - a_{2}) (a_{2} - a_{3}) (a_{3} - a_{1}) (b_{1} - b_{2}) (b_{2} - b_{3}) (b_{3} - b_{1}) < 0

Δ = ∣ ∣ ∣ ∣ \begin{matrix} 1 + a_{1} b_{1} & 1 + a_{1} b_{2} & 1 + a_{1} b_{3} 1 + a_{2} b_{1} & 1 + a_{2} b_{2} & 1 + a_{2} b_{3} 1 + a_{3} b_{1} & 1 + a_{3} b_{2} & 1 + a_{3} b_{3} \end{matrix} ∣ ∣ ∣ ∣

Δ = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \frac{(1 + a_{1} b_{1}) (1 + a_{1}^{2} b_{1}^{2} - a_{1} b_{1})}{1 + a_{1} b_{1}} & \frac{(1 + a_{1} b_{2}) (1 + a_{1}^{2} b_{2}^{2} - a_{1} b_{2})}{1 + a_{1} b_{2}} & \frac{(1 + a_{1} b_{3}) (1 + a_{1}^{2} b_{3}^{2} - a_{1} b_{3})}{1 + a_{1} b_{3}} \frac{(1 + a_{2} b_{1}) (1 + a_{2}^{2} b_{1}^{2} - a_{2} b_{1})}{1 + a_{2} b_{1}} & \frac{(1 + a_{2} b_{2}) (1 + a_{2}^{2} b_{2}^{2} - a_{2} b_{2})}{1 + a_{2} b_{2}} & \frac{(1 + a_{2} b_{3}) (1 + a_{2}^{2} b_{3}^{2} - a_{2} b_{3})}{1 + a_{2} b_{3}} \frac{(1 + a_{3} b_{1}) (1 + a_{3}^{2} b_{1}^{2} - a_{3} b_{1})}{1 + a_{3} b_{1}} & \frac{(1 + a_{3} b_{2}) (1 + a_{2}^{2} b_{2}^{2} - a_{3} b_{2})}{1 + a_{3} b_{2}} & \frac{(1 + a_{3} b_{3}) (1 + a_{3}^{2} b_{3}^{2} - a_{3} b_{3})}{1 + a_{3} b_{3}} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

Δ = 0

Δ

∣ ∣ ∣ ∣ \begin{matrix} x_{1} & y_{1} & 1 x_{2} & y_{2} & 1 x_{3} & y_{3} & 1 \end{matrix} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & 1 a_{2} & b_{2} & 1 a_{3} & b_{3} & 1 \end{matrix} ∣ ∣ ∣ ∣

(x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3})

(a_{1}, b_{1}), (a_{2}, b_{2}), (a_{3}, b_{3})

x = 1 + a + a^{2} + a^{3} + . . .

y = 1 + b + b^{2} + b^{3} + . . . .

1 + a b + a^{2} b^{2} + a^{3} b^{3} + . . . = \frac{x y}{x + y - 1}

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