As we saw previously, a matrix is a rectangular array of elements of ; we say that the matrix is of order if it has m rows and n columns. The set of all matrices of order with entries in is denoted .
The matrix is denoted , or (aij) if no confusion is possible.
If m=n, the matrix is a square matrix of order n.
Let A=(aij) and B=(bij) be two matrices in .
Let A=(aij), B=(bij) and C=(cij) be three matrices in .
The zero matrix: the zero matrix in is the matrix (aij)such that . Multiplication of a matrix by a scalar: Let A=(aij) and B=(bij) be two matrices in and take .
Let , and .
This definition is motivated by the link between matrices and linear operators: the product of two matrices corresponds to the composition of two linear operators (v.s. 1.4).