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Matrix Mathematics (
Posted on 20161030
)
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 20161030 10:07:00
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