Linear Programming and Resource Allocation Modeling

2Mathematical Foundations

2.1 Matrix Algebra

We may define a matrix as an ordered set of elements arranged in a rectangular array of rows and columns. Thus a matrix A may be represented as

where a_ij, i = 1, …, m; j = 1, …, n, is the (representative) element in the ith row and jth column of A. Since there are m rows and n columns in A, the matrix is said to be “of order m by n” (denoted (m × n)). When m = n, the matrix is square and will simply be referred to as being “an nth order” matrix. To economize on notation, we may represent A in the alternative fashion

Oftentimes we shall need to utilize the notion of a matrix within a matrix, i.e. a submatrix is the (k × s) matrix B obtained by deleting all but k rows and s columns of an (m × n) matrix A.

Let us now examine some fundamental matrix operations. Specifically, the sum of two (m × n) matrices A = [a_ij], B = [b_ij] is the (m × n) matrix A + B = C = [c_ij], where c_ij = a_ij + b_ij, i = 1, …, m; j = 1, …, n, i.e, we add corresponding elements. Next, to multiply an (m × n) matrix A by a scalar λ we simply multiply each element of the matrix by the scalar or

(In view of these operations it is evident that A − B = A + (−1) B =  ...