The determinant of a

matrix

is the area of the parallelogram with the column vectors

and

as two of its sides.
Similarly, the determinant of a

matrix

is the volume of the parallelepiped (skew box) with the column vectors

,

, and

as three of its edges.
When the column vectors are linearly dependent, the parallelogram or parallelepiped flattens down at least one dimension and area or volume is zero. Other determinant facts have corresponding geometric interpretations. For instance, doubling a column doubles the area or volume because that doubles one of the dimensions of the parallelogram or parallelepiped.