Determinant and Area
Determinant of a 2 x 2 matrix
Given any 2 x 2 matrix . We know that the determinant . We already learned that if , is invertible. Here we study the determinant from the geometric viewpoint.
We consider the linear transformation such that the matrix for is i.e. for any vector in . In the applet below, you can see how the quadrilateral CDEF is transformed by . Compare the area of the quadrateral before and after the transformation and find out the meaning of .
Suppose the quadrilateral CDEF is a unit square and is any linear transformation defined by you. What is the relationship between and the area of the transformed quadrilateral?
What can you say about the linear transformation when ? Can you give a reason from a geometric viewpoint why is not invertible when ?
Given two 2 x 2 matrices and , it can be shown that . Can you explain why this is true from a geometric viewpoint?