A short method for evaluating determinants and solving systems of linear equations with real or complex coefficients

THE purpose of this paper is to describe a short method for solving arbitrary systems of linear algebraic equations, and evaluating determinants, the quantities involved being either real or complex. For proofs and for treatment of complex systems whose symmetrical coefficients are conjugate, see appendix. The cases considered are: 1. Arbitrary systems with real coefficients, which occur in obtaining stresses in structures, in solving systems of linear differential equations with constant coefficients (transient problems), etc. 2. Symmetrical systems with real coefficients, which occur with d-c networks, undamped vibration, deflections in structures, least square processes, Ritz's method, etc. 3. Symmetrical systems with complex coefficients, which occur with a-c networks, and forced vibration with dissipation. 4. Arbitrary systems with complex coefficients, which occur in certain vibration problems involving gyroscopic action. 5. Systems involving two sets of variables, which occur when the currents in a network are to be found for a variety of impressed voltages, also in the approximate solution of integral equations arising in electric field problems.