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Question

If a1>a2>a3, b1>b2>b3 and aibj1 for 1i, j3, then the determinant
Δ=∣ ∣ ∣ ∣ ∣ ∣1a31b311a1b11a31b321a1b21a31b331a1b31a32b311a2b11a32b321a2b21a32b331a2b31a33b311a1b11a33b321a3b21a33b331a3b3∣ ∣ ∣ ∣ ∣ ∣
is

A
positive
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B
non-negative
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C
negative
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D
non-positive
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Solution

The correct option is A positive
∣ ∣ ∣ ∣ ∣ ∣1a31b311a1b11a31b321a1b21a31b331a1b31a32b311a2b11a32b321a2b21a32b331a2b31a33b311a1b11a33b321a3b21a33b331a3b3∣ ∣ ∣ ∣ ∣ ∣
=∣ ∣ ∣1+a1b1+a21b211+a1b2+a21b221+a1b3+a21b231+a2b1+a22b211+a2b2+a22b221+a2b3+a22b231+a3b1+a23b211+a3b2+a23b221+a3b3+a23b23∣ ∣ ∣
=∣ ∣ ∣1a1a211a2a221a3a23∣ ∣ ∣×∣ ∣ ∣1b1b211b2b221b3b23∣ ∣ ∣
=(a1a2)(a2a3)(a3a1)(b1b2)(b2b3)(b3b1)
This is always positive.

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