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QuesTion 4Decode the following message 29 24 43 39 736073557160by using as an encoding matrix and the numeric representation table that is shown [3 in the following...

Question

QuesTion 4Decode the following message 29 24 43 39 736073557160by using as an encoding matrix and the numeric representation table that is shown [3 in the following figureCharacterA B C D |E | F 6 H| UJ |K L M |NNumeric value2 | 3 |4|5 | 6 | 7 | 8 | 9 10 11 12 13 14CharacterP Q |R |$ iT |U |v |wlY |Y 2SPACENumeric valuel 15 16 17 18 19 20 21 22 23 24 25 26 27

quesTion 4 Decode the following message 29 24 43 39 73 60 73 55 71 60 by using as an encoding matrix and the numeric representation table that is shown [3 in the following figure Character A B C D |E | F 6 H| UJ |K L M |N Numeric value 2 | 3 |4|5 | 6 | 7 | 8 | 9 10 11 12 13 14 Character P Q |R |$ iT |U |v |wlY |Y 2 SPACE Numeric valuel 15 16 17 18 19 20 21 22 23 24 25 26 27



Answers

Refer to the encoding matrix $A=\left[\begin{array}{ll}3 & 5 \\ 1 & 2\end{array}\right]$. The following message was encoded with the matrix $A$. Decode the message. $$ \begin{array}{llllllllll} 31 & 12 & 150 & 55 & 57 & 20 & 150 & 59 & 103 & 39 \end{array} $$ $$ \begin{array}{llllll} 160 & 61 & 61 & 22 & 192 & 73 \end{array} $$

But we're also giving given the may treat Matrix, we're gonna take the inverse of the set that it gives us. gives us 48 18 1:03 41 159, 61, or 22. Sorry, 20- 47 18. one of 5 41. Yeah. And if for you we are if we do the inverse, We end up getting eight size one 20 eight, 27 oh five 475 18. Which ends up being Yeah, Nice space, what you're?

In this problem, you're told that some matrix with a three letter word is multiplied by our including matrix to get are encrypted matrix over here. Now you can multiply this encrypted matrix by the inverse of the encoding matrix to get the original matrix. You could also just saw a fort directly. I'm going to do this, and then we'll ever want to do this is first find the order of individual matrix. Now we see the over here, we have a one by three matrix, and here we have a three by three matrix. Now, this order is comprised of the Commons of the encoding matrix. That's a three. And then the one is the rows of the original matrix. Now, in 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 some numbers A B and C. No. I'm going to multiply this left matrix by this right matrix, looking at the left Forum for Rose in the right for columns and then set that equal to these three numbers. And so when I do this. I get a times one so a minus B. So that's supposed to be times I get a form that's two times see No, that is equal to 40. And then I get one times a So a zero b zero c is equal to five, and then I get zero. A plus two B minus C is equal to negative 17. Now we know what a is so I can play with it into the first equation to solve for BNC, because we'll have two equations with just B and C. And so I get five minus B. Those two C is equal to 40. And so when I saw for that, I get negative. B. Those two C is equal to 35 and then I had B minus. C is equal to negative 17 and I can add the first equation to the second equation to solve for C. And when I do that, I see I get negative. B plus B that zero B two C plus negative CSC and 35 plus negative 17 is 18 so C is equal to 18. Can I want to put this into the second equation here? And I get that be U minus 18 is equal to seven. Negative 17 and so B is equal to home. When we look at our chart, convert our numbers to letters we see that five is equal to e one is equal to an 18 is equal to R and here is our final answer.

In this problem. We're told that we have some original matrix with the word in it multiplied by our encoding matrix. And we get this encrypted matrix. Now we have to determine the size of this additional matrix, and we do that. We're looking at the sizes of the major cities. We do have that This matrix here has one row and three columns. And so it's a one by three matrix. And this matrix here has 123 roofs and 123 columns. It was a three by three matrix. Now, in order to determine the dimensions of the resulting matrix, we take the end. Numbers of the orders of the two matrices were multiplying. And so this three right here is going to be this three. And the first number in the order for this matrix over here is going to be a one. And that's where we get the one over here. And now the second number has to match. Always. The middle numbers always have to match multiplication. So this number here, this three has to match the number of columns in the first matrix. And so our first matrix is going to be a one by three matrix. And so I'm going to write some A B and C in here, and we gonna use those two. So for our word. And so when we do this and we're gonna multiply these two major sees and we have again, we have a resulting one by three matrix, we get the A. But we already know what the numbers in the one by three matrix are. And so I'm going to set the left side equal to the right side. And so I get a plus negative. Beausejour's minus be close to see is equal to 31 right here I get that hang plus zero people. Zero c is equal to eight. Swing on the A musical to eight and I get zero a plus B plus c Hi, politicians. This would actually be a negative one. And so I get zero it plus B minus C. And so I get B minus. C is equal to negative 71. Now I can use a equals eight and put it into the first equation to get two equations with just B and C. And when I do this, I get the eight minus B plus two C is equal to 31 when therefore not gonna be us to see is equal to 23. And then my second equation is being minus. C equals negative 71. And now I'm going to take this first equation and add it to the second equation. And when I do that, I get later B plus B zero b two c plus negative C. It's just see in 23 minus 71 is equal to the following. Actually, I apologize. That should actually just be negative. Seven. There's a typo on the question, and so that didn't miss anything else up. And so when we at 23 to negative seven, we get that C is equal to 16 and I'm going to clothe that into this equation here to sell for B and I see that be minus 16 is equal to negative seven at 16 to both sides, I get that be is equal to nine. And now we know what our A B and two C values are. And if we look in the chart, we see that a a equals eight that's H vehicles. Nine. That's I sequel. 16. That's P R. Answers hip

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.


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