Can we solve aP=b where a and b are each 1 xx n row vectors and are known, and P is an n xx n permutation matrix that is unknown?

Ivan Buckley 2022-09-25 Answered
Can we solve a P = b where a and b are each 1 × n row vectors and are known, and P is an n × n permutation matrix that is unknown?
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Answers (1)

ticotaku86
Answered 2022-09-26 Author has 12 answers
No, It is not possible in general.
Consider P = [ a b c d ]
a = ( 1 , 1 ) and b = ( 2 , 2 )
Now solving
a P = b
amounts to solving the following equation
a + c = 2
b + d = 2
That means a + b + c + d = 4
This is a contradiction as by definition a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere.
So sum of all entries of a n × n permutation matrix is n.
So for a 2 × 2 Permutation matrix sum of all entries is equal to 2.
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