Matrix Multiplication
The row-by-column algorithm that combines matrices in a fundamentally new way.
Introduction
The row-by-column algorithm that combines matrices in a fundamentally new way.
Past Knowledge
You can add matrices and multiply by scalars. You understand the dot product of vectors.
Today's Goal
We multiply matrices using the row-by-column method and understand dimension requirements.
Future Success
Matrix multiplication powers computer graphics (composing transformations), machine learning (neural networks), and physics simulations.
Key Concepts
Dimension Requirement
To multiply , we need (inner dimensions match).
Result dimension: (outer dimensions)
Row-by-Column Algorithm
Entry = (Row of ) · (Column of )
⚠️ NOT Commutative!
In general, . Order matters!
Even when both products exist, they usually give different results.
Worked Examples
Example 1: 2×2 Times 2×2 (Basic)
Compute :
Entry (1,1): Row 1 of A · Column 1 of B
Entry (1,2): Row 1 of A · Column 2 of B
Entry (2,1) and (2,2):
and
Answer:
Example 2: Different Dimensions (Intermediate)
Compute :
Check dimensions: → inner = 3 ✓
Result will be
Entry (1,1):
Entry (2,1):
Answer:
Example 3: Non-Commutativity (Advanced)
Show that :
Compute :
Compute :
Matrix multiplication is NOT commutative!
Common Pitfalls
Multiplying entry-by-entry
Matrix multiplication uses ROW-by-COLUMN dot products, NOT element-wise multiplication!
Ignoring dimension compatibility
For , the number of columns in must equal the number of rows in .
Assuming
Order matters! Always compute in the given order. and are usually different (if both even exist).
Real-World Application
Computer Graphics: Transformation Composition
Each transformation (rotate, scale, translate) is a matrix. To apply multiple transformations, we MULTIPLY the matrices together. The GPU performs billions of matrix multiplications per second to render 3D graphics in real-time video games.
Rotate then scale: (apply right-to-left)
Practice Quiz
Practice Quiz
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