# Geometric interpretation of determinant of a system of homogenous linear equations

What is the geometric interpretation of the determinant of a matrix representing a system of homogenous linear equations?

We know that

• iff the determinant is equal to zero the system has a non-trivial solution and
• in this case the volume of the parallelepiped determined by the column/row vectors of the matrix is also zero (which means that these are linearly dependent).

How can one see the connection between these two facts? How does a volume being zero lead to a non-trivial solution?

• If the volume is zero, the vector taht define the (flat) parallelepiped are not linearly independent. – Bernard Jan 4 '15 at 16:27
• @Bernard: But that is what I wrote: They are linearly dependent?!? – vonjd Jan 4 '15 at 16:29

Here all are homogeneous equations means each one is a subspace of $\mathbb R^k$. So, each line has zero as a point. Hence, these lines are intersecting lines. If two lines are intersecting more than one point, then it must intersect with all other points which gives the determinant is zero.