Mathematical system

The algebra and “mathematical system” theory of matrices is fascinating because it resembles a blend of the way real numbers interact (multiplication and addition exist) with the way vectors interact (addition exists and there’s an “extra” multiplication: scalar multiplication). In addition, matrices can also be interpreted to represent linear transformations of Cartesian n-space, so there’s another layer of meaning involved. In this task, you’ll do some basic exploration of matrices and their transformations of the Cartesian plane.

Requirements:

A. Construct and apply a rotation matrix by doing the following:

1. Create a 2x2 rotation matrix A that is different from I.

2. Determine, showing all work, the location of point (3, 2) when it is rotated using the linear
transformation generated by the matrix A.

B. Construct and analyze a matrix that is not invertible by doing the following:

1. Create a 2x2 matrix B that is not invertible.

2. Demonstrate that matrix B is not invertible.

3. Demonstrate, using B, how to determine
the fourth entry of a matrix that is not invertible when three of the entries are given.

 C. Analyze the invertible matrix M = [ 2 6] by doing the following: 2 4

1. Demonstrate that matrix M is invertible by showing that it has a nonzero determinant.

2. Demonstrate that matrix M is invertible by computing the inverse using the inverse formula for 2x2 matrices.

3. Demonstrate that matrix M is invertible using two additional methods of your choosing.

D. Analyze the linear transformation, L(x) = Dx, that transforms the unit square into the figure below by doing the following:

1. Determine the entries of the 2x2 matrix D.

2. Describe how you determined the entries of D.

3. Compare the values of the determinant of D and the area of the figure, showing all work.