Proof of Cramer's rule with Dirac Delta Function

TL;DR

This paper introduces a novel proof of Cramer's rule using Dirac delta functions, offering a generalized version applicable to partial variable sets, with potential applications in mechanical systems.

Contribution

It provides a new proof of Cramer's rule via Dirac delta functions and derives a generalized rule for partial variable sets, which is a novel contribution.

Findings

Derived a generalized Cramer's rule for partial variables

Presented a new proof of Cramer's rule using Dirac delta functions

Potential applications in changing generalized coordinates in mechanics

Abstract

We present a new proof of Cramer's rule by interpreting a system of linear equations as a transformation of $n$-dimensional Cartesian-coordinate vectors. To find the solution, we carry out the inverse transformation by convolving the original coordinate vector with Dirac delta functions and changing integration variables from the original coordinates to new coordinates. As a byproduct, we derive a generalized version of Cramer's rule that applies to a partial set of variables, which is new to our best knowledge. Our formulation of finding a transformation rule for multi-variable functions shall be particularly useful in changing a partial set of generalized coordinates of a mechanical system.