Solving Cryptographic System 4: Find n Given e & M

[aimfail]

4. Consider a cryptographic system in which a message is list of 5 real numbers: n = [n_1, ... n_5].
The numbers are encrpyted by multiplication with a fixed 5x5 encryption matrix M:
e=[M ]∗n
where "e" is a vector containing the encrypted numbers.
Given
e = [19, 3 , 6 ,11 , 23 ]
and
M=[3 2 1 0 0
0 1 -1 0 1
0 0 1 1 0
1 1 0 1 0
0 5 2 0 -1]
is it possible to find n?
And if yes, find it.

my first thought was just to use row reduction of the matrix:

M=[3 2 1 0 0 19
0 1 -1 0 1 3
0 0 1 1 0 6
1 1 0 1 0 11
0 5 2 0 -1 23]

but a couple of steps later on it just screws up ;(

can anyone help here ?

 

[talolard]

What is the condition for the message to be decrypatble? What did you with M to the original message? What do you need to do to get back to it? How would you do that? How do check if it can be done?