# When a unique solution is found for a matrix of unknown coefficients, A, that have infinite solutions? How to optimise trace(A) s.t. row sum 1?

Let $$\boldsymbol{A}_{(n\times n)}=[a_{ij}]$$ be a square matrix such that the sum of each row is 1 and $$a_{ij}\ge0(i=1,2,\dots,n~\text{and}~j=1,2,\dots,n)$$ are unknown. Suppose that $$\boldsymbol{b}_{1}=[b_{11}~b_{12}\dots b_{1n}]$$ and $$\boldsymbol{b}_{2}=[b_{21}~b_{22}\dots b_{2n}]$$ are known row vectors of proportions such that $$\boldsymbol{b}_{1}\boldsymbol{A}_{(n\times n)}=\boldsymbol{b}_{2},$$ where $$\boldsymbol{b}_{1}\boldsymbol{1}_{n}=1$$, $$\boldsymbol{b}_{2}\boldsymbol{1}_{n}=1$$ and $$\boldsymbol{1}^{T}_{n}=[1~1\dots1]$$.

I know that there are infinite solutions for $$\boldsymbol{A}$$. However, my objective is two-folded: (i) to optimize $$\boldsymbol{A}$$ such that trace$$(\boldsymbol{A})$$ is maximized (and each $$a_{ij}$$ may be expressed in terms of known quantities of the vectors $$\boldsymbol{b}_{1}$$ and $$\boldsymbol{b}_{2}$$ if possible) and (ii) under which condition(s) a unique $$\boldsymbol{A}$$ exists?

Thank you!

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