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Question

If f(y)=∣ ∣ ∣(1y)a1b1(1y)a1b2(1y)a1b3(1y)a2b1(1y)a2b2(1y)a2b3(1y)a3b1(1y)a3b2(1y)a3b3∣ ∣ ∣ and ai,bi are even for i=1,2,3, then f(2) is.

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Solution

f(y)=∣ ∣ ∣(1y)a1b1(1y)a1b2(1y)a1b3(1y)a2b1(1y)a2b2(1y)a2b3(1y)a3b1(1y)a3b2(1y)a3b3∣ ∣ ∣f(y)=∣ ∣ ∣a1b1(1y)a1b11(1y)a1b2(1y)a1b3a2b1(1y)a2b11(1y)a2b2(1y)a2b3a3b1(1y)a3b11(1y)a3b2(1y)a3b3∣ ∣ ∣ +∣ ∣ ∣(1y)a1b1a1b2(1y)a1b21(1y)a1b3(1y)a2b1a2b2(1y)a2b21(1y)a2b3(1y)a3b1a3b2(1y)a3b21(1y)a3b3∣ ∣ ∣ +∣ ∣ ∣(1y)a1b1(1y)a1b2a1b3(1y)a1b31(1y)a2b1(1y)a2b2a2b3(1y)a2b31(1y)a3b1(1y)a3b2a3b3(1y)a3b31∣ ∣ ∣
Putting y=2
We know that ai,bi are even so aibi will also be even.
f(2)=∣ ∣a1b111a2b111a3b111∣ ∣ +∣ ∣1a1b211a2b211a3b21∣ ∣ +∣ ∣11a1b311a2b311a3b3∣ ∣
As two columns are same so,
f(2)=0

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