Determinant and Volume

Determinant of a 3 x 3 matrix

Now let's consider a 3 x 3 matrix

. Suppose

is the linear transformation such that

for any vector

. Analogous to the determinant We define the determinant of

to be the signed volume of the parallelepiped formed by the three vectors

and

. The sign of the determinant is determined by the "right-hand rule" as follows:

Assume

and

are not contained in a plane. Using your right hand, point your index finger in the direction of

and your middle finger in the direction of

as shown in the above figure. If

is pointing in the direction of the thumb, the sign of the

is positive. Otherwise, it is negative. In the applet below, you can see how a unit cube is transformed by

into a parallelepiped. You can also move the unit cube along the grid by dragging the sliders.

What can you say about the column vectors of A when ?