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Matrix Multiplication

Multiplying two matrices combines rows of the first with columns of the second — a very different rule from addition.

Row Times Column

For two 2 \times 2 matrices, each element of the product is found by multiplying a ROW of the first matrix by a COLUMN of the second, and adding the results: (a c b d) \times (e g f h) = (a e + b g ce + d g a f + bh c f + d h)

Matrix lab

Matrix A

Matrix B

AB = [[19,22],[43,50]]

Change a value and watch:

Each entry of the result combines one row of A with one column of B.

Notice the product is NOT found by multiplying matching positions, like addition — it uses whole rows and columns.

Example 1 — multiplying two matrices

Find AB, where A = (1324) and B = (5768) . 1 Top-left: (1 \times 5) + (2 \times 7) = 5 + 14 = 19 2 Top-right: (1 \times 6) + (2 \times 8) = 6 + 16 = 22 3 Bottom-left: (3 \times 5) + (4 \times 7) = 15 + 28 = 43 4 Bottom-right: (3 \times 6) + (4 \times 8) = 18 + 32 = 50 Answer: AB = (19432250)

Example 2 — order matters

Using the same A and B, find BA, and compare with AB. 1 Top-left: (5 \times 1) + (6 \times 3) = 5 + 18 = 23 2 Top-right: (5 \times 2) + (6 \times 4) = 10 + 24 = 34 3 Bottom-left: (7 \times 1) + (8 \times 3) = 7 + 24 = 31 4 Bottom-right: (7 \times 2) + (8 \times 4) = 14 + 32 = 46 Answer: BA = (23313446) — different from AB! Matrix multiplication is NOT commutative: order matters.

Practice

A = (2103) and B = (4012) . Find the element in row 1, column 1 of AB.

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