Evidently, determinantal representation of a solution gives a direct method of its finding analogous to the classical

Cramer's rule that has important theoretical and practical significance [27].

In some cases, the processes of calculation are remarkably similar to other historical gems: for example,

Cramer's rule (Katz, 2004, pp.

Kyrchei obtains explicit formulas for determinantal representations of the W-weighted Drazin inverse solutions (analogs of

Cramer's rule) of the quaternion matrix equations.

It estimates the hazard of chemical structures mainly based on Lipinski's rule and

Cramer's rule. The molecules which are having the hydrogen donors [less than or equal to] 5, hydrogen bond acceptor [less than or equal to] 10, molecular mass [less than or equal to] 500 daltons, and logP [less than or equal to] 5 are likely to obey Lipinski's rule, and

Cramer's rule classifies the chemical compounds into three classes based on the 33 metabolic activities.

The matrix calculation based on

Cramer's rule for a system of equations with three unknowns can be used for simultaneous determination of these compounds in the same ternary mixture and pharmaceutical formulations [13, 26].

Application of

Cramer's rule to (6) yields d[e.sub.m]/d[theta] = ([partial derivative][[GAMMA].sub.1]/[partial derivative][e.sub.s] [partial derivative][[GAMMA].sub.2]/[partial derivative][theta] - [partial derivative][[GAMMA].sub.1]/[partial derivative][theta][partial derivative][[GAMMA].sub.2]/[partial derivative][e.sub.m])/[[DELTA].sub.[GAMMA]], and d[e.sub.s]/d[theta] = ([partial derivative][[GAMMA].sub.1]/[partial derivative][theta][partial derivative][[GAMMA].sub.2]/[partial derivative][e.sub.m] - [partial derivative][[GAMMA].sub.1]/[partial derivative][e.sub.m] [partial derivative][[GAMMA].sub.2]/[partial derivative][theta]/[[DELTA].sub.[GAMMA]].

The map focuses on

Cramer's Rule and its use of determinants to solve a system of n equations.

Special attention is paid to the linear ODE, the operational methods for differential systems of equations (substitution method,

Cramer's rule etc.) to solve algebraic systems, the Laplace transform and the series method for solution about ordinary and regular singular points (Froebenius method [1-3]).

Using

Cramer's rule from (1.1), (2,1), and (6.2) we can solve (3)

There are tables of mathematical relations and of standard component values and a review of

Cramer's Rule for the solution of a set of linear simultaneous equations.

Cramer's rule for the W-weighted Drazin inverse solutions, in particular, has been derived in [27] for singular linear equations and in [26] for a class of restricted matrix equations.

This leads naturally to two other consequences, viz., what is known as Chio's pivotal condensation process for computing determinants and

Cramer's Rule. While the condensation process for computing determinants is known, it is not widely known, and the manner of solving equations developed here has not been seen elsewhere.