Answer
Linearly dependent vectors cannot form a closed region in space so the volume formed with these vectors cannot exist. This means the parallelepiped determined by the three vectors has zero volume.
i.e $det\,\mathbf{A}=Volume \,of\,the\,parallelepiped=0$
Work Step by Step
A Matrix is not invertible if and only if its columns are linearly dependent this means that one of the columns can be written as the linear combination of the other two columns. or simply put, one of the vectors is in the plane spanned by the other two. These vectors cannot form a closed region in space so the volume formed with these vectors cannot exist. This means the parallelepiped determined by the three vectors has zero volume.
i.e $det\,\mathbf{A}=Volume \,of\,the\,parallelepiped=0$
