A message was encoded using the matrix
[7 2
3 1]
and you can decode the message 2 numbers at a time in a [1 x 2] matrix.
Here are the first four numbers:
66 21 119 35
I know it's coded row matrix times the decoder = the uncoded row matrix. (The key is A=1, B=2, etc.) I'm just not sure how to, because i tried it and got really large numbers. Help please?
thank you!
you will need the inverse matrix of
7 2
3 1 which is
1 -2
-3 7
To decode the message using the given matrix [7 2; 3 1], you need to follow these steps:
1. Define the given coded row matrix [66 21 119 35].
2. Split the coded row matrix into individual [1 x 2] matrices. In this case, you should have:
- [66 21]
- [119 35]
3. Set up the matrix equation using the given encoded matrix and a variable matrix for the uncoded row matrix. Let's call the variable matrix [x y]:
[7 2] * [x y] = [66 21]
[3 1] [x y] [119 35]
4. Multiply the matrices on the left side of the equation:
[7x + 2y] = [66 21]
[3x + y] [119 35]
5. Write two linear equations using the values obtained in step 4:
7x + 2y = 66 (equation 1)
3x + y = 21 (equation 2)
Now, we have a system of linear equations with two variables (x and y). We can solve this system of equations to find the values of x and y, which represent the uncoded letters.
To solve the system of equations, you can use various methods such as substitution or elimination. In this case, we will use the elimination method.
First, we'll manipulate equation 2 by multiplying it by 2:
6x + 2y = 42 (equation 3)
Now, subtract equation 3 from equation 1:
7x + 2y - (6x + 2y) = 66 - 42
x = 24
Substitute the value of x back into equation 2:
3(24) + y = 21
72 + y = 21
y = -51
So the decoded letters are x = 24 and y = -51.
However, since the key is A=1, B=2, etc., we need to adjust the values of x and y accordingly. Assuming A is 1, B is 2, and so on, we'll add 64 to each value.
Adjusted value of x = 24 + 64 = 88
Adjusted value of y = -51 + 64 = 13
Therefore, the first two decoded letters are H (88) and M (13).