Matrix Multiplication and What It Represents
A matrix records a linear rule. Multiplying matrices composes the rules, which is why the product is defined by rows against columns and why order matters.
What a learner can do afterwards
- Multiply two matrices and state when the product is defined
- Give two matrices whose products in the two orders differ
- Read a matrix-vector product as a combination of the columns
1 · Read
Think of each matrix as a rule with an input size and an output size. A product AB exists only when the columns of A match the rows of B. Write the shapes side by side, like 2 by 3 next to 3 by 4. The inner pair must agree, and the outer pair gives the answer shape, 2 by 4.
Take A with rows (1, 2) and (3, 4), and B with rows (5, 6) and (7, 8). The top left entry of AB pairs row 1 of A with column 1 of B: 1 times 5 plus 2 times 7, which is 19. Every entry works the same way: one row of A dotted with one column of B.
Order matters, so AB and BA are different beasts. One order may exist while the other fails: a 3 by 2 times a 2 by 5 works, but the reverse fails outright. Even when both orders exist, the values can differ, as with rotations and reflections applied in different orders. Never swap the order silently.
Read a matrix vector product Av as a combination of the columns of A, with the entries of v as weights. Each entry of v scales one column, and you add the scaled columns. As a check, the identity matrix leaves any matching matrix unchanged, much like multiplying by 1.
Check shapes first, pair rows with columns, keep the order, and read Av as weighted columns.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.