dimensions of a matrix
The dimensions of a matrix are just its measurements: how many rows tall and how many columns wide. It is the matrix's “shape,” the same way a photo might be described as 4 by 6.
We write the dimensions as (number of rows) by (number of columns), in that order, and read “m by n.” A matrix with 3 rows and 5 columns is a 3-by-5 matrix; reverse the order and you would be describing a different shape entirely. The total number of entries is m times n.
Dimensions decide what is even allowed. You can add two matrices only when they share the same dimensions, and you can multiply A by B only when A's column count equals B's row count. So before any operation, check the shapes — half of matrix algebra is bookkeeping the dimensions.
[1, 2, 3; 4, 5, 6] has dimensions 2 by 3 (2 rows, 3 columns), so 6 entries. A single column [1; 2; 3] is 3 by 1.
Rows first, columns second.