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, [url=https://www.youtube.com/watch?v=nvHhmEabwqk]this discussion[/url] on calculating determinants by elimination), it can be tricky to understand what it is that you've calculated.[br][br][b]The determinant of a matrix, A, is the volume of the parallelpiped created by the image the standard basis vectors [math]i=(1,0,0)[/math], [math]j=(0,1,0)[/math], [math]k=(0,0,1)[/math] under the linear transformation represented by A. [/b][br][br]In other words, if you perform the matrix multiplications, [math]A*i[/math], [math]A*j[/math], and [math]A*k[/math], the resulting three vectors span a parallelpiped in three dimensions (the 3D version of a parallelogram). The determinant is the volume of this parallelpiped.[br][br]This applet lets you visualize this result. You can adjust the matrix A on the left, and view the parallelpiped spanned by [i]i[/i], [i]j[/i], and [i]k[/i] 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 [url=http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Determinants,%203x3.pdf]here[/url].