Part 1

Now Bob should follow the following steps to decrypt the message

1) First he should arrange the input vector in the form of a 3X3 matrix i.e the first 3 elements form the first column, the next three elements form the second column and so on.
Thus he would get

A= $ \left[ \begin{matrix}1 & 0 & 4 \\ 0 & 1 & 0 \\ 1 & 0 & 1\end{matrix} \right] $

2) Now he should multiply the inverse of the "special matrix" $ M^{-} $ to each column of matrix A to get the respective 3x1 matrices and then combine all three 3x1 matrices to form a new 3x3 matrix
3) Now all he needs to do is replace all the numbers in with their corresponding alphabets and arrange the matrix in the form of a vector.

Eureka!


Part 2

No I don think Eve can decrypt the message without finding out the inverse of the matrix


Part 3

Now the decrypted message is
$ \left[ \begin{matrix}2 \\ 23 \\ 3\end{matrix} \right] $ x $ M^{-} $

$ M^{-} $ = $ \left[ \begin{matrix}1/2 & 0 & 2 \\ 0 & 1 & 0 \\ 1/3 & 0 & 1/3\end{matrix} \right] $

Thus
$ \left[ \begin{matrix}2 \\ 23 \\ 3\end{matrix} \right] $$ M^{-} $ = $ \left[ \begin{matrix}1/2 & 0 & 2 \\ 0 & 1 & 0 \\ 1/3 & 0 & 1/3\end{matrix} \right] $=$ \left[ \begin{matrix}2 \\ 23 \\ 5\end{matrix} \right] $

Replacing the numbers by letters we get "BWE"

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Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett