Here's a list of list of places where life would be miserable w/o matices: 1. Matlab 2. coding theory (Hamming, Huffman, Viterbi, et al.) 3. FTFD simulation 4. BEM (boundary element modeling) 5. FEM ( finite element modeling) 6. Markov processes 7. Hartline-Ratliff models of lateral inhibiton in the retina 8. CFD (computational fluid dynamics) 9. the Jacobian for a truely wide ranging discussion see a book like Applied matrix and tensor analysis by John A. Eisele and Robert M. Mason. New York, Wiley-Interscience  x, 337 p.
At least in the case of Matlab functions like sphere, ndgrid, and meshgrid they can be introduced to and understood by an intelligent 9 year old.