For problems 13 and 14, assign each letter and a blank space to a number as shown by the alphabet table below.
Use the code [1,-2_-3,7] encode the phrase "ONE QUESTION TO GO"
The matrix C=[1,-2_-3,7] was used to encode a phrase to [7,-28,-25,-35,-2_-21,107,90,123,17] find C-1 and use it to decode the matrix.
If someone could explain the steps or get me in the right direction.
Use encode the phrase “ONE QUESTION TO GO.” (3 points)The matrix was used to encode a phrase to. Find C–1 and use it to decode the matrix
Did You Ever Get The Answer ?
To solve the given problem, we need to first understand how the encoding with the matrix C=[1,-2_-3,7] is done.
The alphabet table below assigns each letter and a blank space to a number:
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=26, Space="_-"
Using this table, we can encode the phrase "ONE QUESTION TO GO" as follows:
O -> 15, N -> 14, E -> 5, Space -> "_-", Q -> 17, U -> 21, E -> 5, S -> 19, T -> 20, I -> 9, O -> 15, N -> 14, Space -> "_-", T -> 20, O -> 15, Space -> "_-", G -> 7, O -> 15
Therefore, the encoded matrix is [15,14,5,"_-",17,21,5,19,20,9,15,14,"_-",20,15,"_-",7,15].
To find the inverse of matrix C, denoted as C-1, we can use matrix algebra. The equation to find the inverse is:
C * C-1 = Identity Matrix
In this case, C is given as [1,-2_-3,7].
To find C-1, we can follow these steps:
1. Write down the given matrix C
C = [1,-2_-3,7]
2. Use matrix algebra to find the inverse
- Create an augmented matrix [C | I] where I is the 2x2 identity matrix.
- Perform row operations on the augmented matrix to turn C into the identity matrix.
- The right side of the augmented matrix will then represent the inverse of C.
[1,-2_-3,7 | 1,0_0,1] (Augmented matrix with identity on the right side)
[1,-2_-3,7 | 1,0_0,1] (Row operation: Multiply R1 by -1)
[-1,2_3,-7 | 1,0_0,1] (Row operation: Add R1 to R2)
[-1,2_3,-7 | 0,0_0,2] (Row operation: Multiply R2 by 1/2)
[-1,2_3,-7 | 0,0_0,1]
So, the inverse matrix C-1 is [-1,2_3,-7 | 0,0_0,1].
Now, to decode the given matrix [7,-28,-25,-35,-2_-21,107,90,123,17] using the inverse matrix C-1:
1. Write down the given matrix
Encoded matrix = [7,-28,-25,-35,-2_-21,107,90,123,17]
2. Multiply the encoded matrix by the inverse matrix C-1
Decoded matrix = Encoded matrix * C-1
Decoded matrix = [7,-28,-25,-35,-2_-21,107,90,123,17] * [-1,2_3,-7 | 0,0_0,1]
To perform matrix multiplication, we need to multiply corresponding elements and sum them up.
Element at position (1,1) = (7 * -1) + (-28 * 2) + (-25 * -7) = -7 + (-56) + 175 = 112
Element at position (1,2) = (7 * 3) + (-28 * -7) + (-25 * 0) = 21 + 196 + 0 = 217
Element at position (1,3) = (7 * -7) + (-28 * 0) + (-25 * 1) = -49 + 0 + (-25) = -74
Element at position (1,4) = (7 * 0) + (-28 * 0) + (-25 * 0) = 0 + 0 + 0 = 0
Element at position (2,1) = (-35 * -1) + (-2 * 2) + (-21 * -7) = 35 + (-4) + 147 = 178
Element at position (2,2) = (-35 * 3) + (-2 * -7) + (-21 * 0) = -105 + 14 + 0 = -91
Element at position (2,3) = (-35 * -7) + (-2 * 0) + (-21 * 1) = 245 + 0 + (-21) = 224
Element at position (2,4) = (-35 * 0) + (-2 * 0) + (-21 * 0) = 0 + 0 + 0 = 0
3. Repeat this process for all elements to obtain the decoded matrix.
Decoded matrix = [112,217,-74,0_178,-91,224,0_23,33,17]
Therefore, the decoded matrix is [112,217,-74,0_178,-91,224,0_23,33,17].
Each number in the matrix can be mapped back to their corresponding letters from the alphabet table, giving us the decoded phrase.