System of Equations

Let a linear system of equations be denoted

$${\hbox{\sf A}}{\bf X} = {\bf Y},$$ (1)

where ${\hbox{\sf A}}$ is a Matrix and X and Y are Vectors. As shown by Cramer's Rule, there is a unique solution if ${\hbox{\sf A}}$ has a Matrix Inverse ${\hbox{\sf A}}^{-1}$. In this case,

$${\bf X}={\hbox{\sf A}}^{-1} {\bf Y}.$$ (2)

If ${\bf Y}={\bf0}$, then the solution is ${\bf X}={\bf0}$. If ${\hbox{\sf A}}$ has no Matrix Inverse, then the solution Subspace is either a Line or the Empty Set. If two equations are multiples of each other, solutions are of the form

$${\bf X}={\bf A}+t{\bf B},$$ (3)

for $t$ a Real Number.

See also Cramer's Rule, Matrix Inverse