Adjustment Computations, 6th Edition

APPENDIX BSOLUTION OF EQUATIONS BY MATRIX METHODS

B.1 INTRODUCTION

As stated in Appendix A, an advantage offered by matrix algebra is its adaptability to computer usage. Using matrix algebra, large systems of simultaneous linear equations can be programmed for general computer solution using only a few systematic steps. For example, the simplicities of programming matrix additions and multiplications were presented in Section A.9. To solve a system of equations using matrix methods, it is first necessary to define and compute the inverse matrix.

B.2 INVERSE MATRIX

If a square matrix is nonsingular (its determinant is not zero), it possesses an inverse matrix. When a system of simultaneous linear equations consisting of n equations and involving n unknowns is expressed as AX = B, the coefficient matrix (A) is a square matrix of dimensions n × n. Consider this system of linear equations

(B.1)

The inverse of matrix A, symbolized as A⁻¹, is defined as

(B.2)

where I is the identity matrix. Premultiplying both sides of matrix Equation (B.1) by A⁻¹ gives

Reducing yields

(B.3)