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Given two proper cones $K_1$ and $K_2$ in $\\mathbb{R}^n$ and $\\mathbb{R}^m$, respectively, we say that $A$ is nonnegative if $A(K_1) \\subseteq K_2$. $A$ is said to be semipositive if there exists a $x \\in K_1^\\circ$ such that $Ax \\in K_2^\\circ$. We prove that $A$ is nonnegative if and only if $A+B$ is semipositive for every semipositive matrix $B$. 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