TABLE OF CONTENTS This type of method refers to a procedure for computing a solution from a form that is mathematically exact. We shall begin with a simple method called Cramer s rule with determinants. We shall then continue with the Gaussian elimination method and its variants, methods involving triangular matrices, symmetric and tridiagonal matrices.
This is our first direct method for solving linear systems using determinants. This method is one of the least efficient for solving a large number of linear equations. It is, however, very useful for explaining some problems inherent in the solutions of linear equations.
Consider a system of two linear equations
with the condition that a 11 a 22 ? a 12 a 21 ?0; that is, the determinant of the given matrix must not be equal to zero or the matrix must be nonsingular. Solving the above system using systematic elimination by multiplying the first equation of the system with a 22 and the second equation by a 12, and subtracting, gives
and now solving for x 1, gives
and putting the value of x 1 in any equation of the given system, we have x 2, as follows
Then writing in determinant form, we have
where
In a similar way, one can have Cramer s rule for a set of n linear equations as follows:
that is, the solution for any one of the unknowns x i in a set of...
