Let's prove that matrix multiplication is associative, meaning that for any three matrices A, B, and C, we have:

(AB)C = A(BC)

Step-by-Step Proof:

1. Define Matrices

Let A, B, and C be three matrices of appropriate sizes for multiplication:

- Let A be an m \times n matrix.
- Let B be an n \times p matrix.
- Let C be a p \times q matrix.

Thus, the product AB is an m \times p matrix, and the product BC is an n \times q matrix. Both (AB)C and A(BC) should result in an m \times q matrix.

2. Define Matrix Multiplication

For any matrices X and Y, the (i,j)-th entry of the product XY is:

(XY)_{ij} = \sum_{k} X_{ik} Y_{kj}

Using this formula, we will compute both sides of the associativity equation.

3. Compute (A B) C

First, compute AB:

(AB)_{ij} = \sum_{k} A_{ik} B_{kj}

Now, multiply by C to get (AB)C:

((AB)C)_{ij} = \sum_{r} (AB)_{ir} C_{rj}

Substituting (AB)_{ir} = \sum_{k} A_{ik} B_{kr}, we get:

((AB)C)_{ij} = \sum_{r} \left( \sum_{k} A_{ik} B_{kr} \right) C_{rj}

Rearrange the summation order:

((AB)C)_{ij} = \sum_{k} A_{ik} \left( \sum_{r} B_{kr} C_{rj} \right)

4. Compute A (B C)

First, compute BC:

(BC)_{kj} = \sum_{r} B_{kr} C_{rj}

Now, multiply by A to get A(BC):

(A(BC))_{ij} = \sum_{k} A_{ik} (BC)_{kj}

Substituting (BC)_{kj} = \sum_{r} B_{kr} C_{rj}, we get:

(A(BC))_{ij} = \sum_{k} A_{ik} \left( \sum_{r} B_{kr} C_{rj} \right)

5. Compare Both Sides

We see that:

((AB)C)_{ij} = \sum_{k} A_{ik} \left( \sum_{r} B_{kr} C_{rj} \right)

(A(BC))_{ij} = \sum_{k} A_{ik} \left( \sum_{r} B_{kr} C_{rj} \right)

Since these expressions are identical for all entries i, j, we conclude that:

(AB)C = A(BC)

Thus, matrix multiplication is associative. ✅

ChatGPT usually formats his math-related responses in Markdown with LaTeX math notation. It is possible to ask him to format the responses using code blocks (```math ... ```) for block math and inline math using dollar-backticks ($...$) (KaTeX or MathJax style) - that's what I did to produce the text above. However, then the formatting in ChatGPT itself breaks, however you can copypaste it to Moera.
Shmuel Leib Melamud I wonder what would it take to support LaTeX math notation as well? I mean, what will it mean architecturally? 🙂