Gate_2018_Model_Paper

Question

How to solve it for M^-1?

Comments

M⁴ = I (given)

M³*M=I  =>  M³ *M*M^-1=I*M^-1 => M³=M^-1

Similarly, if M⁴=I then by squaring both sides,  M⁸=I²  =>M⁸= I [ as I²=I] => M⁷=M^-1

Hence we can see that M^-1=M³ or M⁷ or M¹¹...

We can write this as M^-1=M^4k+3  [when k=0 M^-1=M³, k=1 M^-1=M⁷...]

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M^4=I.
(M^4)^2=I.
(M^4)^3=I.
(M^4)^n+1=I.
M^(4n+4)=I.
Now multiply both side by M^-1.
We get,
M^(4n+3)=M^-1.