The Cable and Telecommunications Professionals' Reference, 3rd Edition

22  Matrices and Determinants

J. Barron

Linear Simultaneous Equations

This set of equations:

a₁₁x₁ + a₁₂x₂ + . . . + a_1nx_n = b₁

a₂₁x₁ + a₂₂x₂ + . . . + a_2nx_n = b₂

. . .

a_n1x₁ + a_n2x₂ + . . . + a_nnx_n = b_n

may be written symbolically:

Ax = b

in which A is the matrix of the coefficients a_ij and x and b are the column matrices or vectors (x₁ . . . x_n) and (b₁ . . . b_n). In this case, the matrix A is square (n × n). The equations can be solved unless two or more of them are not independent, in which case:

det A = [A] = 0

and there then exist nonzero solutions x_i only if b = 0. If det A ≠ 0, there exist nonzero solutions only if b ≠ 0. When det A = 0, A is singular.

Matrix Arithmetic

If A and B are both matrices of m rows and n columns, they are