Conditions for a System of Simultaneous Linear Equations to Have Unique Solutions
The simultanous equations
has the unique solution x=3, y=1.
We can write the system in matrix form as
The condition nexcessary for a unique solution to exist is then thatexists, OR, sincethat
We can generalize this.
If a system of linear equations can be associated with a square coeefficient matrix with non zero determinant, then the system has a unique solution. More specifically, if we can write a system of linear equations in the formwhereis a square matrix andis the vector of variables to be solved for, then
Example: Find if the system of equations
has a unique solution, and if so find it.
The third equation is reduncant, since it is twice the second equation. The system is equivalent to the system
The coeefficient matrix iswith determinantso the system has a unique solution, given by