All forms of collaboration are OK, but turn in your final work in your own
handwritten form.

Consider the abstract matrices
A = [1 a 0 0]
    [0 1 0 0]
    [0 b 1 0]
    [0 c 0 1]
and
B = [1 x 0 0]
    [0 1 0 0]
    [0 y 1 0]
    [0 z 0 1]

1. Compute some powers of A, namely
A^0, A^1, A^2, A^3, A^4.

2. Let n be an arbitrary non-negative integer.
Compute A^n.

3. Propose a good definition of the square root A^(1/2) of A, based on question
1 above (in this case MATLAB may give you the good answer).

4. Compute both matrix products A * B and B * A.
What makes the result remarkable (a propery which usually doesn't hold for
matrices)?

5. Consider the abstract matrix with k not equal to 1
C = [1 a 0 0]
    [0 k 0 0]
    [0 b 1 0]
    [0 c 0 1]
Find a nice form for C^n (MATLAB helps, but consider extra simplification).