Does sha256 is ok for that? If not does EC?

I think from what I studied that I had the correct reasoning, but I would like a double check. Here is the thing:

Given two matrix: A and B

I calculate the hashFunction(A, B) = C

Now I calcultate the eigenvalues of B: µ and the associate vector x

Now is the following equation correct:

C * x = hashFunction(A, µ) * x = hashFunction(A, µ*x)

Is it true given any hashfunction? Specifically given sha256, or Elliptic Curve Function, or any other?

EDIT:

My aim is, given a public information, to show that I am the owner of the information and gives other information, and avoids that if someone intercepts the password I give before it comes to the audience, he cannot uses it to gives different information.

So for the problem I gave, data are:

B is public

TO authentify me, I give hash(A, µ) and x

2nd EDIT

The answer to my question seems to rely in Signature as described in this paper. Has anyone an example of signature and verification algorithms ?

3rd EDIT:

So I implemented something, could it be use

encoded_Pwd=str(Input_STR).encode("utf-8")
key=hashlib.md5(encoded_Pwd).hexdigest()
PRIVATE_KEY = base64.urlsafe_b64encode(key.encode("utf-8"))

Then:

PUBLIC_KEY=sha256(PRIVATE_KEY).hexdigest()

I'm not quite sure if I understand your question correctly, but all you need is to generate a Public and Private Keys, then share Public key and Sign your document with your Private key. This way, everybody knowing your Public key will be able to verify, that document is signed by you (i.e. the one who knows Private Key).