Now let's consider a 3 x 3 matrix [math]A[/math]. Suppose [math]T:\mathbb{R}^3\to\mathbb{R}^3[/math] is the linear transformation such that [math]T(x)=Ax[/math] for any vector [math]x[/math] in [math]\mathbb{R}^3[/math]. Analogous to the determinant We define the determinant of [math]A[/math], [math]\det(A)[/math] to be the signed volume of the parallelepiped formed by the three vectors [math]T(\hat{\mathbf{i}}), T(\hat{\mathbf{j}})[/math] and [math]T(\hat{\mathbf{k}})[/math]. The sign of the determinant is determined by the "[b]right-hand rule[/b]" as follows:

Assume [math]T(\hat{\mathbf{i}}),T(\hat{\mathbf{j}})[/math] and [math]T(\hat{\mathbf{k}})[/math] are not contained in a plane. Using your right hand, point your index finger in the direction of [math]T(\hat{\mathbf{i}})[/math] and your middle finger in the direction of [math]T(\hat{\mathbf{j}})[/math] as shown in the above figure. If [math]T(\hat{\mathbf{k}})[/math] is pointing in the direction of the thumb, the sign of the [math]\det(A)[/math] is positive. Otherwise, it is negative.[br][br]In the applet below, you can see how a unit cube is transformed by [math]T[/math] into a parallelepiped. You can also move the unit cube along the grid by dragging the sliders.