Matrices as organised calculations

A matrix preserves two kinds of labels

A stock matrix has rows Shop A, Shop B and columns pens, pads, folders:

S = (2, 1, 3; 1, 4, 2).

Its order is 2 by 3: two rows and three columns. The entry in row 2, column 1 is 1 pen at Shop B. A matrix without its row and column meanings loses the data interpretation.

A delivery with the same labels can be added entry by entry. Matrices can be added only when their orders match and the labels describe compatible quantities. Multiplying S by scalar 2 doubles each stock count.

Matrix multiplication combines corresponding quantities

Suppose unit prices, in dollars, are

p = (5; 8; 2).

The product Sp has order 2 by 1. Its first entry is 2(5) + 1(8) + 3(2) = 24 dollars; its second is 1(5) + 4(8) + 2(2) = 41 dollars. Each row times column pairs matching stock and price labels, then sums the contributions.

For an m × n matrix times an n × k matrix, the result is m × k. The inner dimensions must match. The product pS is not defined here: a 3-by-1 matrix cannot multiply a 2-by-3 matrix because 1 and 2 do not match.

Order can change the result

Even when both products are defined, matrix multiplication is generally not commutative. For

A = (1, 1; 0, 1), B = (1, 0; 1, 1),

AB = (2, 1; 1, 1), BA = (1, 1; 1, 2).

Check one entry at a time. Multiplying matching entries only is not matrix multiplication.

Check your understanding 1 Read dimensions and labels

A 3-by-2 matrix records three customers’ counts of two products. A 2-by-1 column gives the two unit prices. What does their product represent, and what is its order?

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It is a 3-by-1 column of total costs, one for each customer. Each row combines the two purchase counts with their matching prices. The two inner dimensions match.

A numerical result still needs a unit

If stock counts multiply prices, the product entries are monetary totals. If journey counts multiply durations, they are total times instead. Matrix dimensions alone check arithmetic compatibility; labels check whether the calculation makes sense.

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