Question

A and B are two non singular matrices such that A^6 = I and AB^2 = BA (B is not equal to I ). Find the value of A such that B^k = I (where I indicates Identity Matrix)
a)31
b)32
c)64
d)63

Answer 1

The value of A such that B^k = I is K = 63.

Option (D) is correct.

Step-by-step explanation:

A6 = I

A^-1.BA = A^-1.AB^2

A^-1 .BA = B^2

B^2 = A^-1 .BA

B^4 = B^2 . B^2 = (A^-1 .BA) .  (A^-1 .BA) => A^-1B^4.A

B^4 = A^-1 (A^-1 B.A ) A = A^-2 B

B^8 = B^4 . B^4 = (A^-2 B .A^2)  (A^-2 B .A^2)

      => A^-2 B^2.A^2

=> A^-2 A^-1 BAA^2 = A^-3 BA^3

B^16 A^-4 .BA^4

B^32 = A^-5 BA^5

B^64 = A^-6 BA^6

B^64 = IBI

B^64 = B

B^64.B^-1 =  BB^-1 = I

B^63 = I

B^K = I

K = 63

Thus the value of A such that B^k = I is K = 63.