Matrix Multiplication Mechanics

Algebra 2 · Axiom Academy

LESSON Matrix Multiplication Mechanics The row-by-column rule: every entry of AB is the dot product of a row of A with a column of B . Before any arithmetic, check the shapes. Line the two matrices up as and look at the two inner numbers. Once the shapes fit, every entry follows one rule. The entry in row i , column j of AB comes from a single row of A and a single column of B . Take row i of the first matrix. Take column j of the second matrix. Multiply matching entries and add them up. Let's build one specific entry — the one in row 1, column 2 of AB . Pair the three entries of row 1 with the three entries of column 2, then add the products. Row 1 of A is and column 2 of B is . Multiply matching entries, then add: the three products accumulate into a single number for position (1,2) . Now fill the whole product. Each of the four entries of AB uses the same row-by-column rule — one row of A paired with one column of B . 5. Valid vs Invalid Multiplications Whether a product exists is decided entirely by the inner dimensions. Compare two cases built from the same first matrix. Result: undefined — cannot multiply You've built matrix multiplication from a single idea: every entry is a row of A dotted with a column of B . Scroll up to revisit any step.

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