# 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?

Can we solve $\mathbit{a}\mathbit{P}=\mathbit{b}$ where $\mathbit{a}$ and $\mathbit{b}$ are each $1×n$ row vectors and are known, and $\mathbit{P}$ is an $n×n$ permutation matrix that is unknown?
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

ticotaku86
No, It is not possible in general.
Consider $P=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$
$a=\left(1,1\right)$ and $b=\left(2,2\right)$
Now solving
$aP=b$
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.