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Conditions for a System of Simultaneous Linear Equations to Have Unique Solutions

The simultanous equations

x+2y=5

x-y=2

has the unique solution x=3, y=1.

We can write the system in matrix form as

Then

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

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