Question

encode the message math is good break the message into groups of two letters and use the matrix m = \\(\

$$\begin{bmatrix}2 & 7 \\\\ 1 & 3\\end{bmatrix}$$

\\). how can the message be encoded? a translate each group into a sequence of integer pairs and multiply each by m. b translate each group into a sequence of integer pairs and add each to the identity matrix. c translate each group into a sequence of integer pairs and subtract each from the identity matrix. d translate each group into a sequence of integer pairs and multiply each by \\(m^{-1}\\). encode the message. give your answers as column matrices, where each matrix contains a pair of encoded letters. give the matrices in the order they were encoded, which should correspond (type an integer or simplified fraction for each matrix element)

Explanation:

Step1: Assign numbers to letters

First, we assign each letter a number (A=1, B=2, ..., Z=26). The message is "Math is good". Let's break it into groups of two letters (and handle spaces or adjust, but here we'll consider the letters: M(13), A(1), T(20), H(8), I(9), S(19), G(7), O(15), O(15), D(4). Wait, maybe the original message is "Math is good" so let's list the letters with their positions:

• "M" = 13, "A" = 1 → group [13, 1]
• "T" = 20, "H" = 8 → group [20, 8]
• "I" = 9, "S" = 19 → group [9, 19]
• "G" = 7, "O" = 15 → group [7, 15]
• "O" = 15, "D" = 4 → group [15, 4]

Wait, maybe the first part is "Math" (M, A, T, H) → groups [M,A], [T,H]; "is" (I,S) → [I,S]; "good" (G,O,O,D) → [G,O], [O,D]. So the groups are:

1. [13, 1] (M,A)
2. [20, 8] (T,H)
3. [9, 19] (I,S)
4. [7, 15] (G,O)
5. [15, 4] (O,D)

Step2: Multiply each group matrix by M

The matrix \( M =

$$\begin{bmatrix} 2 & 7 \\ 1 & 3 \end{bmatrix}$$

\). For a group matrix \(

$$\begin{bmatrix} a \\ b \end{bmatrix}$$

\), we multiply \( M \times

$$\begin{bmatrix} a \\ b \end{bmatrix}$$

=

$$\begin{bmatrix} 2a + 7b \\ a + 3b \end{bmatrix}$$

\).

Group 1: [13, 1]

Calculate \( 2(13) + 7(1) = 26 + 7 = 33 \) and \( 13 + 3(1) = 13 + 3 = 16 \). So the encoded matrix is \(

$$\begin{bmatrix} 33 \\ 16 \end{bmatrix}$$

\) or as a column matrix (but maybe written as \(

$$\begin{bmatrix} 33 & 16 \end{bmatrix}$$

^T \), but let's present as \(

$$\begin{bmatrix} 33 \\ 16 \end{bmatrix}$$

\) or in the required format. Wait, maybe the problem expects the matrix multiplication as \( M \times

$$\begin{bmatrix} a & b \end{bmatrix}$$

^T \), but let's do each group:

Group 1: \(

$$\begin{bmatrix} 2 & 7 \\ 1 & 3 \end{bmatrix}$$

\times

$$\begin{bmatrix} 13 \\ 1 \end{bmatrix}$$

=

$$\begin{bmatrix} 2*13 + 7*1 \\ 1*13 + 3*1 \end{bmatrix}$$

=

$$\begin{bmatrix} 26 + 7 \\ 13 + 3 \end{bmatrix}$$

=

$$\begin{bmatrix} 33 \\ 16 \end{bmatrix}$$

\)

Group 2: \(

$$\begin{bmatrix} 2 & 7 \\ 1 & 3 \end{bmatrix}$$

\times

$$\begin{bmatrix} 20 \\ 8 \end{bmatrix}$$

=

$$\begin{bmatrix} 2*20 + 7*8 \\ 1*20 + 3*8 \end{bmatrix}$$

=

$$\begin{bmatrix} 40 + 56 \\ 20 + 24 \end{bmatrix}$$

=

$$\begin{bmatrix} 96 \\ 44 \end{bmatrix}$$

\)

Group 3: \(

$$\begin{bmatrix} 2 & 7 \\ 1 & 3 \end{bmatrix}$$

\times

$$\begin{bmatrix} 9 \\ 19 \end{bmatrix}$$

=

$$\begin{bmatrix} 2*9 + 7*19 \\ 1*9 + 3*19 \end{bmatrix}$$

=

$$\begin{bmatrix} 18 + 133 \\ 9 + 57 \end{bmatrix}$$

=

$$\begin{bmatrix} 151 \\ 66 \end{bmatrix}$$

\)

Group 4: \(

$$\begin{bmatrix} 2 & 7 \\ 1 & 3 \end{bmatrix}$$

\times

$$\begin{bmatrix} 7 \\ 15 \end{bmatrix}$$

=

$$\begin{bmatrix} 2*7 + 7*15 \\ 1*7 + 3*15 \end{bmatrix}$$

=

$$\begin{bmatrix} 14 + 105 \\ 7 + 45 \end{bmatrix}$$

=

$$\begin{bmatrix} 119 \\ 52 \end{bmatrix}$$

\)

Group 5: \(

$$\begin{bmatrix} 2 & 7 \\ 1 & 3 \end{bmatrix}$$

\times

$$\begin{bmatrix} 15 \\ 4 \end{bmatrix}$$

=

$$\begin{bmatrix} 2*15 + 7*4 \\ 1*15 + 3*4 \end{bmatrix}$$

=

$$\begin{bmatrix} 30 + 28 \\ 15 + 12 \end{bmatrix}$$

=

$$\begin{bmatrix} 58 \\ 27 \end{bmatrix}$$

\)

Wait, but maybe the initial grouping is different. Let's check the message "Math is good" – maybe "Math" is M(13), A(1), T(20), H(8); "is" I(9), S(19); "good" G(7), O(15), O(15), D(4). So groups of two: (13,1), (20,8), (9,19), (7,15), (15,4). Then multiplying each by M as above.

Answer:

The encoded matrices (column matrices) are:

\(

$$\begin{bmatrix} 33 \\ 16 \end{bmatrix}$$

\), \(

$$\begin{bmatrix} 96 \\ 44 \end{bmatrix}$$

\), \(

$$\begin{bmatrix} 151 \\ 66 \end{bmatrix}$$

\), \(

$$\begin{bmatrix} 119 \\ 52 \end{bmatrix}$$

\), \(

$$\begin{bmatrix} 58 \\ 27 \end{bmatrix}$$

\)

(If the problem expects the matrices in a different format, adjust accordingly, but based on the steps, these are the encoded matrices by multiplying each group matrix by \( M \).)