Order and entries

A matrix with m rows and n columns has order m × n, read 'm by n'. Rows are always stated first.

An m × n matrix holds mn entries. A 3 × 2 matrix has 6; a 3 × 3 matrix has 9.

The entry in row i and column j is written aᵢⱼ, and the whole matrix as A = [aᵢⱼ] of order m × n, with 1 ≤ i ≤ m and 1 ≤ j ≤ n.

Possible orders for a given number of entries come from its factor pairs. A matrix with 8 entries can be 1 × 8, 8 × 1, 2 × 4 or 4 × 2; one with a prime number of entries, such as 13, can only be 1 × 13 or 13 × 1.

A rule for aᵢⱼ builds a matrix entry by entry. For a 3 × 2 matrix with aᵢⱼ = ½|i − 3j|: a₁₁ = 1, a₁₂ = 5/2, a₂₁ = 1/2, a₂₂ = 2, a₃₁ = 0, a₃₂ = 3/2.

Counting: if every entry of a 3 × 3 matrix is 0 or 1, each of the 9 entries has 2 choices, so there are 2⁹ = 512 such matrices.