AP® Precalculus · Unit 4 · Topic 4.10
Matrix Multiplication
Matrix multiplication combines each row of the first matrix with each column of the second using dot products, producing a new matrix whose entries measure how one transformation acts through another.
Matrices represent transformations of the plane, so multiplying them should mean doing one transformation after another. That is why entries don't just match up position by position like addition does. Instead, the entry in row i, column j of the product comes from taking row i of the first matrix and column j of the second matrix, multiplying matching entries together, and adding the results — a dot product. This pairing captures how every output direction of the second transformation gets fed into the first.
Each dot product needs the row's length to match the column's length, so the number of columns in the first matrix must equal the number of rows in the second. A 2x2 matrix times a 2x2 matrix always works, since every row has 2 entries and every column has 2 entries. The resulting matrix takes its number of rows from the first matrix and its number of columns from the second.
Because AB pairs A's rows with B's columns, while BA pairs B's rows with A's columns, the two products generally land on completely different numbers — matrix multiplication is not commutative. This sits right next to matrix transformations: multiplying two matrices is the same as applying one geometric transformation then another, and the order you apply them in changes where points end up, just like multiplying the matrices in a different order changes the product.
Key terms
- Dot product
- The sum of the products of corresponding entries in a row and a column.
- Matrix product
- A new matrix whose entries are dot products of the first matrix's rows with the second matrix's columns.
- Commutative
- A property where order doesn't change the result; matrix multiplication generally lacks this property.
- Compatible dimensions
- The requirement that the number of columns in the first matrix equal the number of rows in the second for multiplication to be defined.
Watch it explained
- How it worksHow does each entry of the product come from a dot product of one row with one column?
- Worked exampleFind the product of the matrices with rows (2, 1) and (0, 3) times rows (1, -1) and (4, 2).
- Common mistakeWhy isn't matrix multiplication the same in both orders?
- Going furtherWhat has to be true about the two shapes before matrices can even be multiplied?
Check yourself
End of topic.