For the initiated, this is SICP[1] Exercise 1.19. That exercise walks you through a proof without explicitly using linear algebra. I remember having a blast solving this back in the day.
[1] https://web.mit.edu/6.001/6.037/sicp.pdf => page 61
[1] https://web.mit.edu/6.001/6.037/sicp.pdf => page 61
Makes me want to sit and go through the book.
Thanks for sharing. :)