When Can We Multiply?

Linear Algebra (Matrices) · Axiom Academy

The dimension rule for matrix multiplication: the inner dimensions must match. 1. The Inner Dimensions Must Match Tag each matrix with its shape and slide them next to each other: for A , then for B . The two numbers that end up touching in the middle are the columns of A and the rows of B . They must be the same single number n — that is the entire rule. Inner numbers match ( n = n ) → the product is defined The result takes its shape from the outer numbers 2. Why the Inner Number Has To Agree The rule isn't arbitrary. Each entry of is built by pairing one row of A with one column of B , multiplying matching entries and adding. For every entry to find a partner, the row and the column must be the same length — and that length is exactly (columns of A ) on one side and (rows of B ) on the other. Has one entry per column of A — so its length is n , the column count of A . Has one entry per row of B — so its length is also n , the row count of B . Equal lengths means every entry meets a partner: multiply, then sum into one number. If the lengths differ, an entry is left with no partner — the sum is undefined. "Columns of A equals rows of B " is just "the row and the column are the same length so they can be dotted." That single shared length is the inner number n . 3. When Multiplication Is Undefined

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.