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 2×2 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 2×2 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 2×2 matrices.
3. Demonstrate that matrix M is invertible using two additional methods of your
choosing.
