question on code-breaking?

Question

The following coded-message was encoded with a 2x2 matrix.
"8, 21, -15, -10, -13, -13, 5, 10, 5, 25, 5, 19, -1, 6, 20, 40, -18, -18, 1, 16"
If you chanced upon the information that the last word of the message is actually "_RON", can you derive the pre-specified
martix?

A is represented by number 1, B by 2, C by 3, etc, and finally "_" by 0.

Answer 1

In order to decode the message, we interpret each pair of encoded numbers as a vector. There are 10 vectors:

<8, 21>, <- 15, - 10>, <- 13, - 13>, <5, 10>, <5, 25>, <5, 19>, <- 1, 6>, <20, 40>, <- 18, - 18>, <1, 16>

Since _RON corresponds to the vectors <0, 18>, <15, 14>, we know that:

M<0, 18> = <-18, -18>
M<15, 14> = <1, 16>

We know that for any 2×2 matrix M, the first column of M is M<1, 0> and the second is M<0, 1>. Thus, we need to find M<1, 0>. Fortunately, M is a linear transformation, so if there are constants a and b such that a<0, 18> + b<15, 14> = <1, 0>, then:

M<1, 0> = M(a<0, 18> + b<15, 14>) = aM<0, 18> + bM<15, 14> = a<-18, -18> + b<1, 16>

To find a and b, we equate x and y components of the vectors, giving the system of equations:

15b=1
18a+14b=0

Which has the solution:

a = -7/135, b = 1/15

So M<1, 0> = -7/135<-18, -18> + 1/15<1, 16> = <1, 2>

Now to find M<0, 1>. First we solve:

15b=0
18a+14b=1

The solution is:

a=1/18, d=0

So <0, 1> = 1/18<0, 18>, and:

M<0, 1> = 1/18<-18, -18> = <-1, -1>

Thus M is the matrix:

[1, -1]
[2, -1]

To decode the message, we multiply every vector by M^-1, which is:

[-1, 1]
[-2, 1]

The decoded vectors are:

<13, 5>, <5, 20>, <0, 13>, <5, 0>, <20, 15>, <14, 9>, <7, 8>, <20, 0>, <0, 18>, <15, 14>

Which spells out the message:

MEET_ME_TONIGHT__RON