Tropical algebras are semirings with operations $a\oplus b=\max(a,b)$ (or $a\oplus b=\min(a,b)$) and $a\otimes b=a+b$. Tropical algebra is usually approached as a subtopic of algebraic geometry. But what if we take traditional linear algebraic problems and ask them using tropical operations instead? We'll find that these problems are frequently solvable, although not always in the same ways as they're solved in conventional algebras. These problems have surprising relevance to industrial systems control and operations research problems.