Assume $A$ is a $n\times n$ matrix where $A=B+E$, $E$ is an identity matrix. Prove:
If $B$ is a positive definite matrix, then $Ax=F$ has unique solution.
Assume $A$ is a $n\times n$ matrix where $A=B+E$, $E$ is an identity matrix. Prove:
If $B$ is a positive definite matrix, then $Ax=F$ has unique solution.
Suppose there are two solutions $x$ and $y$ such that $Ax=Ay=F$, then take the difference: $A(x-y)=0$, but $A$ is positively definite, hence, $x=y$.