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Matrix Mathematics (Posted on 2016-10-30) Difficulty: 3 of 5
Suppose A and B are two non singular matrices such that AB = BA^2 and B^5 = I, then prove that A^31= I

Note: Here I is the identity matrix.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution Solution Comment 1 of 1
Let B' denote the inverse of B.  Then B^5=I implies B'^5=I.

Rearrange A*B=B*A^2 into 
A = B*A^2*B'

Then substitute the latter equation into itself to get 
A = B^2*A^4*B'^2

Substitute three more times to get the series of equations
A = B^3*A^8*B'^3
A = B^4*A^16*B'^4
A = B^5*A^32*B'^5

The last one simplifies to I = A^31.

  Posted by Brian Smith on 2016-10-30 10:07:00
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