Let A=[anbn]
=[√3+11−√3√3−11+√3][an−1bn−1]
∵sinπ12=√3−12√2, cosπ12=√3+12√2
A=2√2⎡⎢
⎢⎣cosπ12−sinπ12sinπ12cosπ12⎤⎥
⎥⎦[an−1bn−1]
=(2√2)2⎡⎢
⎢⎣cosπ12−sinπ12sinπ12cosπ12⎤⎥
⎥⎦2[an−2bn−2]
...
=(2√2)n−1⎡⎢
⎢⎣cosπ12−sinπ12sinπ12cosπ12⎤⎥
⎥⎦n−1[a1b1]
∴[a22b22]=(2√2)21⎡⎢
⎢⎣cosπ12−sinπ12sinπ12cosπ12⎤⎥
⎥⎦21[11]
We know that,
[cosx−sinxsinxcosx]n=[cosnx−sinnxsinnxcosnx]
∴[a22b22]=(2√2)21⎡⎢
⎢⎣cos7π4−sin7π4sin7π4cos7π4⎤⎥
⎥⎦[11]
=(2√2)21⎡⎢
⎢
⎢
⎢⎣1√21√2−1√21√2⎤⎥
⎥
⎥
⎥⎦[11]
=231[11−11][11]
=232[10]
Hence, a22=232⇒n=32