Find seq. of elem. matrices Ek,...,E1 so Ek...E1A = I for A=

[maogdamian]

Question: mostly irrelevant, but if you're interested--find a sequence of elementary matrices Ek,...,E1 such that Ek...E1A=I for a matrix A = {{2, 4}, {1, 1}} [that matrix is burned in my skull right now]. Write A and A^-1 in terms of elementary matrices.

Answer: Choose one of the following. I've kept a log of either A or A^-1; the other can be easily computed visually since these elementary matrices are very easy to invert. Dot means matrix multiplication, it's Mathematica notation.

1. {{0, 1}, {1, 0}} . {{1, 0}, {2, 1}} . {{1, 0}, {0, 2}} . {{1, 1}, {0, 1}}
2. Inverse[{{1, 0}, {0, -1/2}} . {{1, 3/2}, {0, 1}} . {{1, 0}, {-1, 1}}.{{1, -1}, {0, 1}}]
3. Inverse[{{1, 0}, {0, -1}}. {{1, 2}, {0, 1}}.{{1, 0}, {-1, 1}}.{{1/2, 0}, {0, 1}}]
4. {{2, 0}, {0, 1}}.{{1, 0}, {1, 1}}.{{1, -2}, {0, 1}}.{{1, 0}, {0, -1}}
5. Inverse[{{1, 0}, {-1, 1}}.{{-1/2, 0}, {0, 1}}.{{1, -4}, {0, 1}}]
6. {{2, 0}, {0, 1}}.{{1, 0}, {1, -1}}.{{1, 2}, {0, 1}}
7. Inverse[{{1, 0}, {0, 1/2}}.{{1, -1/2}, {0, 1}}.{{1, 0}, {-2, 1}}.{{0, 1}, {1, 0}}]
8. Inverse[{{0, 1}, {1, 0}}.{{1, 0}, {-1, 1}}.{{1/2, 0}, {0, 1}}.{{1, -2}, {0, 1}}]
9. Inverse[{{1, -1}, {0, 1}}.{{1, 0}, {-1, 1}}.{{1, 0}, {0, 1/2}}.{{0, 1}, {1, 0}}]
10. {{1, 1}, {0, 1}}.{{1, 0}, {1, 1}}.{{1, 0}, {0, -2}}.{{1, 3}, {0, 1}}
11. Inverse[{{1, -2}, {0, 1}}.{{1, 0}, {1, -1}}.{{1/2, 0}, {0, 1}}]
12. Inverse[{{1, -1}, {0, 1}}.{{1, 0}, {0, 1/2}}.{{0, 1}, {1, 0}}.{{1, -2}, {0, 1}}]
13. Inverse[{{0, 1}, {1, 0}}. {{1, 0}, {-1, 1}}.{{1, -1}, {0, 1}}.{{1/2, 0}, {0, 1}}]
14. {{2, 0}, {0, 1}}.{{1, 0}, {1, 1}}.{{1, 0}, {0, -1}}.{{1, 2}, {0, 1}}
15. Inverse[{{1, -1}, {0, 1}}.{{1, 0}, {0, 1/2}}.{{1, 0}, {-2, 1}}.{{0, 1}, {1, 0}}]
16. Inverse[{{1, -2}, {0, 1}}.{{1, 0}, {0, -1}}.{{1, 0}, {-1, 1}}.{{1/2, 0}, {0, 1}}]
17. Inverse[{{1, -1}, {0, 1}}.{{1, 0}, {0, 1/2}}.{{1, 0}, {-2, 1}}.{{0, 1}, {1, 0}}]
18. {{2, 0}, {0, 1}}.{{0, 1}, {1, 0}}.{{1, 0}, {1, 1}}.{{1, 1}, {0, 1}}
19. Inverse[{{-1, 0}, {0, 1}}.{{1, 0}, {1, 1}}.{{1, -2}, {0, 1}}.{{1/2, 0}, {0, 1}}]
20. {{1, -1}, {0, 1}}.{{1, 0}, {-1, 1}}.{{1, 0}, {0, -1/2}}.{{0, 1}, {1, 0}}
21. Inverse[{{1, -3}, {0, 1}}.{{1, 0}, {0, -1/2}}.{{1, 0}, {-1, 1}}.{{1, -1}, {0, 1}}]
22. Inverse[{{1, -1}, {0, 1}}.{{0, 1}, {1, 0}}.{{1, -1}, {0, 1}}.{{1/2, 0}, {0, 1}}]
23. Inverse[{{1, 0}, {0, -1/2}}.{{1, 1}, {0, 1}}.{{1, 0}, {0, 2}}.{{1, 0}, {-1, 1}}.{{1/2, 0}, {0, 1}}]



Yes, the answer to EVERY ONE of the ~120 copies of this problem is one of the above; they are *all* equivalent, and elementary. Except that I'm only half done, so I'm sure there's maybe half a dozen new ones I'll find. And this is just the *correct* answers.