20260412

Below is a reminder of some MATLAB experiments NOT for points. The problems for
points are farther below.

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Use R2022b MATLAB or later.

(1) Test whether symbolic MATLAB is installed. Start MATLAB and do commands:

a = [1 2 ; 3 4]
b = [4 ; 5]
syms x y
eqn = a * [x ; y] == b
solu = solve([eqn], [x, y])
solu.x
solu.y

(2) For bigger square matrices we may define:

badA = [1 2 3 ; 4 5 6 ; 7 8 9]
det(badA)
badA^(-1)
A = sym([1 2 3 ; 4 5 6 ; 7 8 9])
and B = sym([6 ; 15 ; 24])
det(A)
A^(-1)
syms x y z
eqn = A * [x ; y ; z] == B
solu = solve([eqn], [x, y, z])
solu.x
solu.y
solu.z

(3) In (2) still only one solution [0;3;0], MATLAB ignores kernel [1;-2;1]
HOWEVER, there is the nullspace operator and the range operator
null(A) (gives orthogonal basis)
colspace(A) (gives orthogonal basis)
So try:

null(A)
colspace(A)

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Problems for points. Beware that we want EXACT answers, no rounding. So 1/3 is
good, but 0.3333333333 is not.

All forms of collaboration are OK, but turn in your final work in your own
handwritten form (some students hand-write comments on printouts, OK too).

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. Let A^t and B^t be the transposes of A and B respectively (even google knows
how to write the transpose of a matrix in MATLAB).
Compute A * B^t and B * A^t.
Subjective question: Do you see differences and/or similarities
between the two matrix products, and can you explain them?

6. 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, consider simplifications
to make the final answer look better).

We precede problem 7 below with a little explanation.
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, for example a fixed k satisfying 0 <= k <= 3, we can
write nchoosek(n,k) as a simple polynomial in n (although
MATLAB usually doesn't).

7. Consider abstract matrix
A = [a 1 0 0]
    [0 a 1 0]
    [0 0 a 1]
    [0 0 0 a]
(We state without proof: A is not diagonalizable.)
(a) Compute A^1 = A, A^2 = A*A, A^3 = A*A*A, and A^4 = A*A*A*A.
(b) Give the correct formula for A^n, where n is an
arbitrary positive integer.

8. Compute the null space N(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]

9. 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.