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? 🙂