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Question

A=[1tanxtanx1] and f(x) is defined as f(x)=det.(ATA1) then find the value of f(f(f(f.......f(x))))n times is (n2).

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Solution

Given A=[1tanxtanx1]
|A|=1+tan2x
AT=[1tanxtanx1]
Here,c11=1,c12=tanxc21=tanx,c22=1
adjA=[1tanxtanx1]T=[1tanxtanx1]
A1=11+tan2x[1tanxtanx1]
Now, ATA1=11+tan2x[1tanxtanx1]=[1tanxtanx1]
ATA1=11+tan2x[1tan2x2tanx2tanx1tan2x]
ATA1=[cos2xsin2xsin2xcos2x]
|ATA1|=sin2x+cos2x=1
Given ,f(x)=det.(ATA1)
f(x)=1
f(f(f(f.......f(x))))n times=1

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