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Matrix Mathematics (Posted on 2016-10-30)

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

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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