Matrix Multiplication Visualization: $AB$

Overview

The animation demonstrates the step‑by‑step computation of the product of two $2\times2$ matrices $A = \begin{pmatrix}1 & 2\\ 8 & -1\end{pmatrix}$ and $B = \begin{pmatrix}3 & 4\\ 5 & 6\end{pmatrix}$. Viewers see each row of $A$ paired with each column of $B$, the element‑wise products highlighted, summed, and the resulting entry placed into the matrix $AB$. The key takeaway is how matrix multiplication is performed via dot products of rows and columns.

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Phases

#	Phase Name	Duration	Description
1	Intro	~2s	Show the two original matrices $A$ and $B$ side‑by‑side with a placeholder matrix $AB = \bigl(\ \bigr)$ beneath them.
2	Compute $c_{11}$	~6s	Row 1 of $A$ and Column 1 of $B$ separate, column transposes, element‑wise products shown with “+”, sum placed into the top‑left entry of $AB$.
3	Compute $c_{12}$	~6s	Row 1 of $A$ with Column 2 of $B$ undergo the same process, result placed in top‑right entry.
4	Compute $c_{21}$	~6s	Row 2 of $A$ with Column 1 of $B$, result placed in bottom‑left entry.
5	Compute $c_{22}$	~6s	Row 2 of $A$ with Column 2 of $B$, result placed in bottom‑right entry.
6	Reveal Full Result	~3s	The completed matrix $AB$ is highlighted, optionally showing the full numeric matrix $\begin{pmatrix}13 & 16\\ 19 & 26\end{pmatrix}$.

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Layout

┌─────────────────────────────────────────────┐
│                     MAIN                     │
│  (matrices A, B, placeholder AB, animated      │
│   rows/columns, intermediate products)       │
├─────────────────────────────────────────────┤
│               Caption (optional)             │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Main	Matrices $A$ and $B$ displayed left/right, placeholder $AB$ below; animated rows, columns, product terms, and the growing entries of $AB$.	Uses simple rectangles for matrix cells; numbers are shown as LaTeX text inside cells.
Caption	Small label indicating the current entry being computed, e.g., "Computing $c_{11}$".	Omitted if the animation is self‑explanatory; kept brief.

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Notes

• Assumptions: No persistent narration; all information conveyed visually. The animation uses a single Scene class, with each phase sequenced linearly.
• The placeholder matrix is shown as $AB = \bigl(\ \bigr)$ (empty parentheses) until entries are filled.
• Row/column separation is animated slowly (≈1 s for the move, ≈1 s for the transpose, ≈1 s for each product term, ≈1 s for the sum) to satisfy the request for a “slow” demonstration.
• After each entry is placed, the matrix cell briefly flashes or changes color to emphasize the update.
• The final result matrix is displayed for a few seconds before the scene ends.
• No additional text is used beyond the optional caption; all mathematical expressions are rendered with LaTeX.