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

Look up `binomial coefficient' on (for example) Wikipedia.
Using google to search for `binomial coefficient' gives you
many links like the one for Wikipedia.
MATLAB uses the function nchoosek for binomial coefficients.
So nchoosek(5,2) returns 10.
For small k, like a fixed k satisfying 0 <= k <= 3, you can
write nchoosek(n,k) as a simple polynomial in n (although
MATLAB doesn't).

Consider abstract matrix
A = [a 1 0 0]
    [0 a 1 0]
    [0 0 a 1]
    [0 0 0 a]

1. Compute A^1 = A, A^2 = A*A, A^3 = A*A*A, and A^4 =
A*A*A*A.

2. Give the correct formula for A^n, where n is an
arbitrary positive integer.

In MATLAB, the null space N(B) of a matrix B is the span of
null(B), and the rangle R(B) of a matrix B is the span of
colspace(B).

3. and 4. Compute the null space N(B) of abstract matrix B
and the range R(B) of abstract matrix B, where
B = [0 1 0 0]
    [0 a 1 0]
    [0 0 a 1]
    [0 0 0 0]
    [0 0 0 b]

For this final homework, you have to look up with MATLAB
help how to compute eigenvalues and eigenvectors. Experiment
a bit first, for example with easy tiny 2x2 matrices.

5. Check that the matrix A above problem 1 can never be
diagonalizable. This can even be shown without MATLAB.

6. Find eigenvalues and eigenvectors of abstract matrix
A = [4   1 -1]
    [4   0  2]
    [2  -1  3]
Write your resulting diagonal matrix C as a product as in
equation C = P^(-1) * A * P.