Determinant and Area

Determinant of a 2 x 2 matrix
Given any 2 x 2 matrix [math]A=\begin{pmatrix}a & b\\c & d\end{pmatrix}[/math]. We know that the determinant [math]\det(A)=ad-bc[/math]. We already learned that if [math]\det(A)\ne 0[/math], [math]A[/math] is invertible. Here we study the determinant from the geometric viewpoint.[br][br]We consider the linear transformation [math]T:\mathbb{R}^2\to\mathbb{R}^2[/math] such that the matrix for [math]T[/math] is [math]A[/math] i.e. [math]T(x)=Ax[/math] for any vector [math]x[/math] in [math]\mathbb{R}^2[/math]. In the applet below, you can see how the quadrilateral CDEF is transformed by [math]T[/math]. Compare the area of the quadrateral before and after the transformation and find out the meaning of [math]\det(A)[/math].[br][br]
Suppose the quadrilateral CDEF is a unit square and [math]T[/math] is any linear transformation defined by you. What is the relationship between [math]\det(A)[/math] and the area of the transformed quadrilateral?
What can you say about the linear transformation [math]T[/math] when [math]\det(A)=0[/math]? Can you give a reason from a geometric viewpoint why [math]A[/math] is not invertible when [math]\det(A)=0[/math]?
Given two 2 x 2 matrices [math]A[/math] and [math]B[/math], it can be shown that [math]\det(AB)=\det(A)\det(B)[/math]. Can you explain why this is true from a geometric viewpoint?
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– Ku, Yin Bon (Albert)

Information: Determinant and Area