Matrix Multiplication

Chapter 2 • Section 2-3

"It's not just multiplying numbers... it's a row-column dance! 💃"

✖️ The Product $AB$

Row-by-Column Rule

The entry in row $i$ and column $j$ of the product $AB$ is the dot product of:

  • Row $i$ of $A$
  • Column $j$ of $B$
$$ (AB)_{ij} = \sum_{k=1}^n a_{ik}b_{kj} $$

Compatibility Rule

$m \times \mathbf{n}$
×
$\mathbf{n} \times p$
=
$m \times p$

Inner dimensions must match!

"The number of columns in the first matrix must equal the number of rows in the second."

📏 Size Matters Win $10

Can you multiply a $2 \times 3$ matrix by a $2 \times 3$ matrix?

⚠️ Order Matters!

In general, $AB \neq BA$

Matrix multiplication is NOT commutative.

Case 1: $AB$ exists, but $BA$ is undefined (different sizes).
Case 2: Both exist, but have different sizes.
Case 3: Both exist and are same size, but entries differ.
Rare Case: They are equal (we say they commute).

Properties that DO Hold

Associativity

$A(BC) = (AB)C$

Parentheses don't matter.

Distributivity

$A(B+C) = AB + AC$

Multiplication distributes over addition.

Identity

$AI = A$ and $IA = A$

Multiplying by Identity does nothing.

Transpose of Product

$(AB)^T = B^T A^T$

Warning: The order flips!

🧠 Logic Check Win $20

If $A$ is $2 \times 3$ and $B$ is $3 \times 4$, what is the size of $(AB)^T$?

Hint: $AB$ is $2 \times 4$, so its transpose is...