Visualizing 3X3 Determinants
One of the trickiest concepts to understand in Linear Algebra is the determinant of a matrix. While it is a straightforward algorithm to compute a determinant (see, for example, this discussion on calculating determinants by elimination), it can be tricky to understand what it is that you've calculated.
The determinant of a matrix, A, is the volume of the parallelpiped created by the image the standard basis vectors , , under the linear transformation represented by A.
In other words, if you perform the matrix multiplications, , , and , the resulting three vectors span a parallelpiped in three dimensions (the 3D version of a parallelogram). The determinant is the volume of this parallelpiped.
This applet lets you visualize this result. You can adjust the matrix A on the left, and view the parallelpiped spanned by i, j, and k on the right. The determinant is automatically calculated for you below A as you adjust the matrix.
Try visualizing the determinants from some of the practice problems here.