Understand multiplication as dot products and composition
Matrix multiplication is not entrywise multiplication. It is designed so that matrices representing linear transformations compose correctly. The (i,j) entry of AB is the dot product of row i of A with column j of B.
The dimension rule
Determine whether a product exists before touching the entries
The number of columns of the left matrix must equal the number of rows of the right matrix. These matching values are the inner dimensions; the remaining outer dimensions become the shape of the product.
A shape
Two output rows and three input coordinates.
B shape
Three rows match A’s three columns.
AB shape
The product retains A’s row count and B’s column count.
Row-by-column computation
Build each output entry as a dot product
Multiply a 2×3 matrix by a 3×2 matrix
| 1 | 2 | −1 |
| 0 | 3 | 4 |
| 2 | 1 |
| −1 | 3 |
| 4 | 0 |
| −4 | 7 |
| 13 | 9 |
For example, c₁₂=(1)(1)+(2)(3)+(−1)(0)=7. Each output entry consumes one complete row from A and one complete column from B.
Three complementary interpretations
Move beyond the mechanical algorithm
Row-column dot products
Best for hand computation. Entry cᵢⱼ measures how row i of A interacts with column j of B.
Columns of AB
Column j of AB is A times column j of B. Therefore each output column is a linear combination of the columns of A.
Composition of maps
If B acts first and A acts second, then the combined transformation is AB. The rightmost matrix acts first.
AB and BA are different questions
Even when both products exist, AB generally differs from BA because the transformations occur in opposite orders. In many rectangular cases, one order exists while the other does not.
Interactive multiplication laboratory
Change the dimensions and inspect every dot product
Variable-size matrix multiplier
Choose m, n, and p for (m×n)(n×p). The calculator displays the complete row-by-column expansion.
Properties and limits
Know which familiar scalar rules survive
| Property | Status | Meaning |
|---|---|---|
| A(BC)=(AB)C | Always valid when dimensions fit | Associativity permits regrouping without changing order. |
| A(B+C)=AB+AC | Valid | Multiplication distributes over addition. |
| AB=BA | Generally false | Matrix multiplication is noncommutative. |
| AB=0 ⇒ A=0 or B=0 | False | Nonzero matrices can multiply to the zero matrix. |
| (AB)ᵀ=BᵀAᵀ | Valid | Transpose reverses composition order. |
| (AB)⁻¹=B⁻¹A⁻¹ | Valid when both are invertible | The last-applied transformation must be undone first. |
Review questions
Test dimensions, computation, and interpretation