# Conditions for uniqueness of solution to a linear system of equation

Consider a $n\times n$ M-Matrix $\mathbf{A}$ and a $n\times n$ non-negative and non-zero matrix $\mathbf{B}$. Also, let $\mathbf{x}$ and $\mathbf{b}$ be two (non-zero) n-column vectors. I am looking for conditions on $\mathbf{B}$ which guarantee the existence of a unique solution to the linear system of

$$(\mathbf{A}+\mathbf{B})\mathbf{x} = \mathbf{b}$$

Some notes: The linear system of $\mathbf{A}\mathbf{x} = \mathbf{b}$ has a unique solution as M-matrix $\mathbf{A}$ is always non-singular (this is a property of non-singular matrices). But I don't know which conditions the non-negative and non-zero matrix $\mathbf{B}$ should satisfy to guarantee the existence of the unique solution to the linear system outlined above. Any comments or solution would be appreciated.

• If $A$ is non-singular, then a possible condition on $B$ is $B=0$. – Dietrich Burde Jun 6 '16 at 18:37
• Of Course, but I am looking for something more applicable and general. – abari Jun 6 '16 at 18:42
• Do you know anything else about $A$? Is it diagonally dominant, by any chance? – parsiad Jun 6 '16 at 19:29
• @par: Yes, $\mathbf{A}$ is diagonally dominant. Can I use this for more general conditions on $\mathbf{B}$? – abari Jun 6 '16 at 21:43

Since $A$ is a nonsingular M-matrix, the simplest condition you can impose is that $B$ is a nonnegative diagonal matrix (so that $A+B$ is nonsingular).