Computing a Matrix Product

Linear Algebra (Matrices) · Axiom Academy

EXAMPLE Computing a Matrix Product Multiply two matrices step by step — each entry is the dot product of a row of A with a column of B . Compute the matrix product AB , where Nice work. You built the product AB one entry at a time, each as a row-column dot product. Row-column rule: Entry (i,j) of AB comes from row i of A and column j of B : . Dot product: Multiply matching terms and add — for 3 -wide rows, a_1b_1 + a_2b_2 + a_3b_3 . One per position: A product needs 9 dot products, worked independently entry by entry. Dimensions must match: AB exists only when A 's columns equal B 's rows; if A is and B is , then AB is . The (1,1) entry pairs row 1 of A (blue) with column 1 of B (orange): (2)(1) + (0)(3) + (1)(2) = 4 . This row-column pattern is the heart of matrix multiplication — it powers transformations in computer graphics, systems of equations, data science, and physics.

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