Mathematical Modeling in the Social and Life Sciences

APPENDIX II

Matrices

By a matrix, we simply mean a rectangular array of numbers. Examples of matrices are

Matrices are classified according to their size and shape by specifying the number of their rows and columns. An m × n matrix is a matrix with m rows and n columns. Thus, A is a 2 × 3 matrix, B is a 2 × 2 matrix, C is a 1 × 4 matrix, and D is a 5 × 1 matrix. Any 1 × n matrix is called a row vector, while an m × 1 matrix is said to be a column vector. The individual numbers in vectors are called components of the vector. If m = n, the matrix is said to be square and to have an order equal to the number of rows. Thus, B is a square matrix of order 2. Two matrices have the same size if they have the same number of rows and columns. The four matrices A, B, C, and D, it should be noted, are of different sizes.

The general m × n matrix M has the form

The number in the ith row and jth column of a matrix is called the ijth entry of the matrix, and may be denoted M_ij. Thus, A₂₃ = 7.

Two matrices are said to be equal matrices if they are of the same size and the corresponding entries are all equal. We write A = B to denote that two matrices A and B are equal. For example, if we consider the six matrices:

only the matrices U and W are equal.

Matrix addition is defined for matrices of the ...