A matrix is a rectangular array of numbers arranged in rows and columns (the plural of matrix is matrices). For example, consider the income and age of 3 people. begin{equation*} mathbf{A}= begin{array}{c} Row~1 \ Row~2 \ Row~3 end{array} overset{ begin{array}{cc} ~~~Col~1~ & Col~2 end{array} }{left( begin{array}{cc} 6,000 & 23 \ 13,000 & 47 \ 11,000 & 35 end{array} right) } end{equation*}
Here, column 1 represents income and column 2 represents age. Each row corresponds to an individual. For example, the first individual is 23 years old with an income of $6,000.
The number of rows and columns is called the dimension of the matrix. For example, the dimension of the matrix (mathbf{A}) above is (3times 2) (read 3 “by” 2). This stands for 3 rows and 2 columns. If we were to represent the income and age of 100 people, then the dimension of the matrix would be (100times 2).
It is convenient to represent a matrix using the notation begin{equation*} mathbf{A}=left( begin{array}{cc} a_{11} & a_{12} \ a_{21} & a_{22} \ a_{31} & a_{31} end{array} right) . end{equation*} Here, (a_{ij}) is the symbol for the number in the (i)th row and (j)th column of (mathbf{A}). In general, we work with matrices of the form begin{equation*} mathbf{A}=left( begin{array}{cccc} a_{11} & a_{12} & cdots & a_{1k} \ vdots & vdots & ddots & vdots \ a_{n1} & a_{n2} & cdots & a_{nk} end{array} right) . end{equation*} In this case, the matrix (mathbf{A}) has dimension (ntimes k).
A vector is a special matrix. A row vector is a matrix containing only 1 row ((k=1)). A column vector is a matrix containing only 1 column ((n=1)). For example, begin{equation*} text{column vector}rightarrow left( begin{array}{c} 2 \ 3 \ 4 \ 5 \ 6 end{array} right) ~~~~text{row vector}rightarrow left( begin{array}{ccccc} 2 & 3 & 4 & 5 & 6 end{array} right) . end{equation*} Notice above that the row vector takes much less room on a printed page than the corresponding column vector. A basic operation that relates these two quantities is the transpose. The transpose of a matrix (mathbf{A}) is defined by interchanging the rows and columns and is denoted by (mathbf{A }^{prime }) (or (mathbf{A}^{T})). For example,
begin{equation*} mathbf{A}=left( begin{array}{cc} 6,000 & 23 \ 13,000 & 47 \ 11,000 & 35 end{array} right) ~~~mathbf{A}^{prime }=left( begin{array}{ccc} 6,000 & 13,000 & 11,000 \ 23 & 47 & 35 end{array} right) . end{equation*} Thus, if (mathbf{A}) has dimension (ntimes k), then (mathbf{A}^{prime }) has dimensions (ktimes n).