Question 53634This question is from textbook elementary linear algebra
: the follow cryptogram was encoded with a 2 X 2 matrix 5,2,25,11,-2,-7,-15,32,14,-8,-13,38,19,-19,-19,37,16
the last word of the message is -Sue what is the message?
I am not sure what 2 by 2 matrix to use?
This question is from textbook elementary linear algebra
Answer by AnlytcPhil(1806) (Show Source):
You can put this solution on YOUR website! the follow cryptogram was encoded with a 2 X 2 matrix
5,2,25,11,-2,-7,-15,32,14,-8,-13,38,19,-19,-19,37,16
the last word of the message is -Sue what is the message?
I am not sure what 2 by 2 matrix to use?
--------------------------------------------------------
You accidentally left out a number in the message and
that caused me to have to go to a lot of trouble to
discover what number you left out. But I finally
found it. You left out the number -30 between the
-15 and the 32.
You had this:
5,2,25,11,-2,-7,-15,32,14,-8,-13,38,19,-19,-19,37,16
and it should have been this:
5,2,25,11,-2,-7,-15,-30,32,14,-8,-13,38,19,-19,-19,37,16
Make that as this list of 1 by 2 matrices:
[5 2],[25 11],[-2 -7],[-15 -30],[32 14],[-8 -13],[38 19],[-19 -19],[37 16]
Let a blank space "_" be 0, let A be 1, B be 2, etc. In other
words:
_=0, A=1, B=2, C=3, D=4, E=5, F=6, G=7, H=8, I=9, J=10, K=11,
L=12, M=13, N=14, O=15, P=16, Q=17, R=18, S=19, T=20, U=21,
V=22, W=23, X=24, Y=25, Z=2600
Let the decoding matrix (the inverse of the coding matrix) be
[a b]
[ ]
[c d]
Don't confuse the small letters a,b,c,d with the capital letters
A,B,C,D as they are different.
Since _=0, S=19, U=21, and E=5
The last two 1 by 2 matrices in the
coded message
[-19 -19], [37 16]
must decode as the numbers corresponding to
[- S],[U E]
which are
[0 19],[21 5]
So [19, -19]must decode to [_ S], or [0, 19]
Therefore,
[a b]
[-19 -19]·[ ] = [0 19]
[c d]
Multiplying the matrices on the left:
[-19a-19c -19b-19d] = [0 19]
So we have the equations
-19a-19c = 0
-19b-19d = 19
Also 37, 16 must decode to U, E, or 21, 5
[a b]
[37 16]·[ ] = [21 5]
[c d]
Multiplying the matrices on the left:
[37a+16c 37b+16d] = [21 5]
So we have the equations
37a+16c = 21
37b+16d = 5
So we have the four equations:
-19a-19c = 0
-19b-19d = 19
37a+16c = 21
37b+16d = 5
Solving the 1st and 3rd
-19a-19c = 0
37a+16c = 21
that gives a=1 and c=-1
Solving the 2nd and 4th
-19b-19d = 19
37b+16d = 5
that gives b=1 and d=-2
So the DEcoding matrix is
[a b] [1 1]
[ ] = [ ]
[c d] [-1 -2]
So we start decoding
[5 2],[25 11],[-2 -7],[-15 -30],[32 14],[-8 -13],[38 19],[-19 -19],[37 16]
Decoding the [5 2]
[1 1]
[5 2][ ] = [5-2 5-4] = [3 1] = [C A]
[-1 -2]
Decoding the [25 11]
[1 1]
[25 11][ ] = [25-11 25-22] = [14 3] = [N C]
[-1 -2]
Decoding the [-2 -7]
[1 1]
[-2 -7][ ] = [-2+7 -2+14] = [5 12] = [E L]
[-1 -2]
Decoding the [-15 -30]
[1 1]
[-15 -15][ ] = [-15+15 -15+30] = [0 15] = [_ O]
[-1 -2]
Decoding the [32 14]
[1 1]
[32 14][ ] = [32-14 32-28] = [18 4] = [R D]
[-1 -2]
Decoding the [-8 13]
[1 1]
[-8 -13][ ] = [-8+13 -8+26] = [5 18] = [E R]
[-1 -2]
Decoding the [38 19]
[1 1]
[38 19][ ] = [38-19 38-38] = [19 0] = [S _]
[-1 -2]
And of course [-19 -19] decodes as [_ S] and
[37 16] decodes as [U E}
so the decoded message is
[C A},{N C],[E L],[_ O],{R D],[E R],[S _],[_ S], [U E]
CANCEL_ORDERS_ _SUE
Edwin
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