Notation
A. Matrix/vector/scalar
-
A matrix may be designated by a single symbol, such as the letter A. Frequently capital letters are used to designate matrices, while lower case letters are used to describe vectors and scalars. So, we can write, for example,
or 
or 
.
B. Cell Entries
-
Cell entries of a matrix may be designated by a symbol with two subscripts. The first subscript indicates the row location and the second subscript indicates the column location of the cell. Frequently lower case letters are used to indicate cell entries of matrices. For any matrix designated by a capital letter, the corresponding lower case letter is used for the cell entries. For example, the cell entries in the matrix A are designated by A
ij
, where
i
indicates the row location and
j
indicates the column location. So, for example, I can write:
for the matrix 
.
C. Order
-
The order of a matrix refers to the size of the matrix and is simply the number of rows and number of columns of the matrix. Thus, a matrix with m rows and n columns is of order m by n or m x n. For example, the matrix
is a 3 x 3 matrix; 
is a 3 x 1 matrix (also sometimes called a column vector of length 3). 
would be, I guess, technically a 1 x 1 matrix as well as a scalar.
D. Mathematical operations
-
Just as in scalar operations, +, -, and * are used to designate addition, subtraction, and multiplication. A superscipt T or diacritical mark is used to designate transposes (described below), and a superscript -1 is used to indicate inverses, the analogous operation to division.