Abstract
This chapter helps one gain a basic understanding and proficiency in the manipulation of linear equations. Matrix techniques are applied for solving linear equations and for determining characteristic equations, roots, and vectors. For single solution systems of the form Ay=b, each solution for y can be obtained by determinant procedures. However, if the system has many variables, solving the system by matrix inversion procedures may be more practical. Elementary operations can be used to find solutions when matrix A is either invertible or noninvertible and when the system is either homogeneous or nonhomogeneous. The characteristic equation of a system of linear equations is a polynomial of degree n in the λi characteristic roots, which is obtained by expanding determinant (A - λI) = 0. Then, equation (A - λI)y = 0 must hold true for each of the characteristic roots. Corresponding to each characteristic root, there is a characteristic vector vi such that (A- λiI) = 0.
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