Formula for Homogeneous System of Linear Equations
Determining whether a homogeneous linear system possesses a unique solution (trivial) or an infinite number of solutions (nontrivial) involves examining the determinant of the coefficient matrix. If A represents the coefficient matrix of the system, then:
The system has a unique solution (trivial) if [Tex]\text{det}(A) \neq 0[/Tex].
The system has an infinite number of solutions (nontrivial) if det(A)=0.
For example, let’s calculate the determinant of the coefficient matrix of a system examined in the previous section:
[Tex]\left| \begin{array}{ccc} 1 & 1 & 1 \\ 0 & 1 & -1 \\ 1 & 2 & 0 \end{array} \right|[/Tex]
= 1 · (0 + 2) – 1 · (0 + 1) + 1 · (0 – 1)
= 2 – 1 – 1
= 0
As the determinant equals 0, the system possesses an infinite number of solutions.
Homogeneous Linear Equations
Linear algebra serves as the backbone for various mathematical concepts, from computer graphics to economic modeling. One fundamental aspect of linear algebra is solving systems of linear equations. Among these, homogeneous systems of linear equations hold particular significance due to their unique properties and applications across diverse fields. In this article, we will understand homogeneous systems, explore their characteristics, methods of solution, and real-world implications.