# Condition for a unique solution of matrix A that have infinite solutions and optimisation of tr(A)

#### zeeshas901

##### New member
Hello!

I am new here, and I need (urgent) help regarding the following question:

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, I do not have any idea about the following two questions: (i) how 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) subject to the sum of each row of$\boldsymbol{A}$is 1 and (ii) under which condition(s) a unique$\boldsymbol{A}\$ exists?

Thank you!