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Iaolycn (Y,E) b }uo Lel (X,5) ~ 6 Leck Hln proe Apue (XsY;€) be K aes preyecol T {XrY _Y T XxY _ X 7 (w < 9 T(x,)) = K A= eon Uhuoty hnekia f I_ IR Ane p" slynomta( J Cudean 4 Y~ Contin1x Exc € Se '~J Gf c{f( I" "Joalk 'w #,6 4i" 0, IR" 19 Dcl 'PaoA (6776) Y~6 rck ILnele Wu f con(i Ihul * mr ~(ostd cel 517 losed (R 8"(940) . $ ILI % '-o," A)c 6of w y"r Sow Deline ( ,Ir CR k3 (*y ele Ukal 0 Lninlow Iah rekavk V (-^1)0 ( ow ) spen Io ip 62 0" (v) ~ptn



Answers

Apply the following decoding scheme: $$\begin{array}{|c|c|c|c|c|c|} \hline 1 & \mathrm{A} & 10 & \mathrm{J} & 19 & \mathrm{S} \\ \hline 2 & \mathrm{B} & 11 & \mathrm{K} & 20 & \mathrm{T} \\ \hline 3 & \mathrm{C} & 12 & \mathrm{L} & 21 & \mathrm{U} \\ \hline 4 & \mathrm{D} & 13 & \mathrm{M} & 22 & \mathrm{V} \\ \hline 5 & \mathrm{E} & 14 & \mathrm{N} & 23 & \mathrm{W} \\ \hline 6 & \mathrm{F} & 15 & \mathrm{O} & 24 & \mathrm{X} \\ \hline 7 & \mathrm{G} & 16 & \mathrm{P} & 25 & \mathrm{Y} \\ \hline 8 & \mathrm{H} & 17 & \mathrm{Q} & 26 & \mathrm{Z} \\ \hline 9 & \mathrm{I} & 18 & \mathrm{R} & & \\ \hline \end{array}$$ The encoding matrix is $\left[\begin{array}{rrr}1 & 1 & 0 \\ -1 & 0 & 1 \\ 2 & 0 & -1\end{array}\right]$. The encrypted matrices are given below. For each of the following. determine the 3-letter word that is originally transmitted. Hint: All six words are parts of the body.$$\text { Cryptography. }[40 \quad 5 \quad-17]$$

In this problem were given these shown encoding matrix and the shown encrypted matrix until that some matrixes multiplied by the encoding matrix. To get this encrypted matrix. Now you are are able to multiply the encrypted matrix by the inverse of the encoding matrix. In order to find out what the original three word or three that are word waas. However, you can also solve for the components individually in the way that we're going to do. This is first determined. The order of the original Matrix. Now, as you can see, are encrypted. Matrix is a one by three matrix, and our recording matrix is also er is a three by three matrix. Now, the order of the product matrix is made up of the calm number of the encoding matrix in the road number of the original matrix and a matrix multiplication. The middle numbers have to match, so this number has to equal this number. And so we originally had a one by three matrix. And so I'm going to write that there were some numbers A, B and C, and here and then I'm going to multiply this matrix by this major extensive. It equal to this matrix. And so when I do this, I'm gonna look at the last number. Are the left matrix furrows in the everyone for columns? Someone to take a times? One. Let's be times negative. Be so I get a minus bi. Those two C is equal to negative 10 and then I get a zero. B zero C is equal to five, and then I get zero. A plus B minus C is equal to 20. Now. We've already solved for what A Is someone a. Plug this into the first equation to give us two equations with B and C only. And so I get five minus B plus two C is equal to negative 10 or negative B plus to see is equal to negative 15. And then I also get B minus. C is equal to 20 and want to add the first equation to the second equation. And when I do this, I get native peoples be that zero B two C plus negative CSC. They get a 15 plus 20 is five, so I got that C is equal to five, and then and then I get B minus five is equal to 20 and I get that B is equal to 25. If we look at our conversion of numbers to letters, we see that five is equal to E 25 people to why and again five people to e. Under answer is I.

And this problem, we are told that some matrix over here is multiplied by the encoding matrix, and the result is the encrypted matrix Over here. Now you can multiply the in verse of the encoding matrix by the encrypted matrix. However, you can also try to solve for the numbers in the first matrix, which is what I am going to do not to do this. I have to know the orders of each matrix. I see that this matrix over here is a one by three matrix. This matrix here is a three by three matrix. Now product major cities, they're always made up of the numbers that are the same as the outer numbers to meet, line up the two orders of the two. Major sees me multiply. So this number here equals this number, and this number is equal to the number of rows of this matrix over here. And then when we multiply, the middle numbers have to be equal. So this number has to equal this number. And so our first matrix is a one by three matrix. Now I'm going to call that some matrix a B C. And now I'm going to multiply the two major sees and set that equal to these three numbers. And so I want to look at the left matrix for Rose and the right for Combs. And so when I do this, I get a tense one, which is a Cosby Times negative one negative B plus C times, too. So plus, to see that's equal to nine. And then I get a plus zero. B zero C is equal to one so is equal to one. And then I get a zero times A with zero A plus. B minus C is equal to five now from a equals one. I can use that to plug into the first equation to make two equations with just B and C. When I do that, I get one minus B. Those two C is equal to nine or negative B that's to see is equal to eight and we have be minus. C is equal to five. I'm going to add the first equation to the second equation, and when I do that, I get negative people. Speed zero b two C plus negative. See that See eight plus five is 13. We get C is equal to 13. I'm going to plug that into this second equation here. And I see that B minus 13 secrets of five and therefore B is equal to 18. Looking at our conversion of numbers to letters, we see that one stands for a 18 stands for our and 13 stands for M, and we found our word.

Okay, So for this one, we're told the s some encrypted messages that were encrypted using these cipher P plus 10 mod 26. Now, this was the encryption cipher. So to decrypt it, we need to undo that. Right? So we're going to use the decryption cipher P minus 10 mod 26. All right, so for a are given message is see e be be Oh, eggs an OBE, x y and G. OK, so first we need to convert that into numeric ce, where a call zero equals one Sequels to case of sea is going to ese for 11 14 23 13 14 and one, 23 24 and six. So now we could run it through our algorithm where we take the number. It's a track 10. And then take my 26. Right, So two months, 10 being negative. Eight mon 26 which is really 18. All right, so now well, cover into those numbers. So we get 18. Same thing here in native six month 26. That would be 20. Okay, then we get 17 17 four, 13 3 4 17 13 14 and 22. And then We just need to translate that back into the alphabet. So it's gonna be an S u R r e and D he r n o w spooky. It says surrender now it's probably how some of this homework makes you feel. Okay, So, onto part B here were given the phrase 00 W i p be s oh X. And so first, we want to translate this into the numbers numerals using the equal zero equals one Sequels too. Okay, so we get that Ella's 11. Oh, is 14 w is 22 eyes, eight p is 15 be is one s is 18 0 is 14 x is 23 it is 13 and I we're gonna use our cipher, right. Which recall is I'm going to be p minus 10 March 26. Yes. So when we take 11 minus 10 while 26 is could be the one 14 minus 10 is four 12 b negative. Tuesdays will be 24 five. That's negative. Nines. Women with 17 eighties gonna be 8 14 4 23 is 13 and 13 is three then Now we have to translate it back, and we do that we get Be? Yeah. And why f o r i e and d be my friend a little bit nicer than the last one. All right, so now we have the last one. Here we are, given the phrase d s. I should probably leave more space than this. A d s w Ow p Why be p e x? So again we translate it into the numbers using a zero beak Was one Sequels too? S o D is gonna be three s is 18 w is 22 0 is 14 a. P is 15. Why is 24 b is one? Why is 15 eat is four and X is 23. And then we put a cipher on it, which was P minus 10 mile 26 to get to the next stage. And when we do that. So this would be negative sevens that ends up giving us a 19. This will be an eight 12 four five, 14 1 becomes negative. Nine. That is a 17 five negative six is gonna be 20 and a 13. And then we translated back into letters, so ah, this time we get tea. I e f o r s u n time for fun

And this problem we're told. So we have an encoding matrix and encrypted matrix, and we want some matrix here to determine what that is in order to determine what are three letter word is now, in order to determine the size of that matrix, we look at the size of these two matrices. Now we can see that this is a three by three matrix and this is going to be a one by three matrix. Now, the resulting matrix of idiot product is always the dimensions on the outer. There's always the outer numbers of the two orders of the major cities were multiplying. And so because this is a one by three matrix and this is a three here, we know that this matrix is going to have one row and then we know and we know that when we multiply matrices, the middle numbers have to match. So this number has to match this number here. And so this is going to be a one by three matrix. And so we're going to have some number A, some number B and some number See in this matrix that we're going to multiply by this matrix in order to get this one. And so when we do that, we see done the first row times. The first column we get a minus. B plus to see is equal to this number here. So if you thought and then we also get that be, this is needed that a I zero b plus zero C is equal to 10. And finally, we know that eight times 00 a plus B minus C is equal to negative 22. No, because we know that A is equal to 10. We can plug that in into this equation here in order to get an equation with just being see Then then we can solve in a system of equations with the third equation. And so when we do that, we get 10 minus B plus two. Si is equal to 55 and so we get negative. B Plus two C is equal to 45 and they also have that B minus. C is equal to negative 22. And as I am this first equation to the second equation, someone added to the second equation. I get zero B plus C because I have two C minus C is equal to 45 minus 22 is 23. So I have that c is equal to 23. And so I've solved for my A over here and my see here. Now I need a sulfur be and I'm going to put it into this equation here, and I get B minus 23. It is equal to negative 20 to add 23 to both sides. I see that B is equal to one. Now I have my values for a B and C, and I'm going to look in the chart to determine what they are. And when I look in the chart, I see that 10 sense for J that one that stands for a and that 23 u stands for W. And so my hidden word was jaw.


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