Part C: Application of linearity
1. How can Bob decrypt the message?
In order to decrypt the message, Bob must call upon his linear algebra skillz. He has been given the 3-by-3 special decoder matrix that he will be able to multiply the inverse of to the coded message matrix he received via email. The result will be the decoded message.
Visually,
$ \begin{bmatrix} Coded Matrix \end{bmatrix} \cdot \begin{bmatrix} Secret Matrix \end{bmatrix} $ -1 = $ \begin{bmatrix} Uncoded Matrix \end{bmatrix} $
For Eve to crack the code, all she needs to do is take the inverse of the input matrix and multiply it by the output matrix. This will give her the (Until Recently) Secret Matrix with which she can start her career at NSA.
Again, visually,
$ \begin{bmatrix} Uncoded Matrix \end{bmatrix} $ -1
$ \cdot \begin{bmatrix} Uncoded Matrix \end{bmatrix} $
=
$ \begin{bmatrix} Secret Matrix \end{bmatrix} $
