Answer to Question #232437 in Linear Algebra for Ash

Question

Use a single 3x3 matrix A =[ 1 4 3 ]to encode the message “THE COAST IS CLEAR”.

[-2 1 5 ]

[ 2 -1 -4 ]

Answer 1

This message,  “THE COAST IS CLEAR”, should be presented as a sequence of numbers, each of them is the number of a letter in the alphabet, 27 is a whitespace. Then form a matrix from them and multiply by the coding matrix given in the task. The first matrix should have 3 columns to be multiplied by the matrix 3x3, and 6 rows, because the message have 18 symbols, 18/3=6.

$$\begin{pmatrix} 20&8&5 \\ 27&3&15 \\ 1&19&20 \\ 27&9&19 \\ 27&3&12 \\ 5&1&18 \\ \end{pmatrix} * \begin{pmatrix} 1 & 4 & 3\\ -2 & 1 & 5 \\ 2 & -1 & -4 \end{pmatrix} = \begin{pmatrix} 14&83&80 \\ 51&96&36\\ 3&3&18\\ 47&98&50\\ 45&99&48\\ 39&3&-52 \end{pmatrix}$$

The components of the matrix are calculated this way:

$$c11 = a11 · b11 + a12 · b21 + a13 · b31 =$$

$$= 20 · 1 + 8 · (-2) + 5 · 2 = 20 - 16 + 10 = 14$$

$$c12 = a11 · b12 + a12 · b22 + a13 · b32 =$$

$$=20 · 4 + 8 · 1 + 5 · (-1) = 80 + 8 - 5 = 83$$

$$c13 = a11 · b13 + a12 · b23 + a13 · b33 =$$

$$= 20 · 3 + 8 · 5 + 5 · (-4) = 60 + 40 - 20 = 80$$

$$c21 = a21 · b11 + a22 · b21 + a23 · b31 =$$

$$= 27 · 1 + 3 · (-2) + 15 · 2 = 27 - 6 + 30 = 51$$

$$c22 = a21 · b12 + a22 · b22 + a23 · b32 =$$

$$= 27 · 4 + 3 · 1 + 15 · (-1) = 108 + 3 - 15 = 96$$

c23 = a21 · b13 + a22 · b23 + a23 · b33 = 27 · 3 + 3 · 5 + 15 · (-4) = 81 + 15 - 60 = 36

c31 = a31 · b11 + a32 · b21 + a33 · b31 = 1 · 1 + 19 · (-2) + 20 · 2 = 1 - 38 + 40 = 3

c32 = a31 · b12 + a32 · b22 + a33 · b32 = 1 · 4 + 19 · 1 + 20 · (-1) = 4 + 19 - 20 = 3

c33 = a31 · b13 + a32 · b23 + a33 · b33 = 1 · 3 + 19 · 5 + 20 · (-4) = 3 + 95 - 80 = 18

c41 = a41 · b11 + a42 · b21 + a43 · b31 = 27 · 1 + 9 · (-2) + 19 · 2 = 27 - 18 + 38 = 47

c42 = a41 · b12 + a42 · b22 + a43 · b32 = 27 · 4 + 9 · 1 + 19 · (-1) = 108 + 9 - 19 = 98

c43 = a41 · b13 + a42 · b23 + a43 · b33 = 27 · 3 + 9 · 5 + 19 · (-4) = 81 + 45 - 76 = 50

c51 = a51 · b11 + a52 · b21 + a53 · b31 = 27 · 1 + 3 · (-2) + 12 · 2 = 27 - 6 + 24 = 45

c52 = a51 · b12 + a52 · b22 + a53 · b32 = 27 · 4 + 3 · 1 + 12 · (-1) = 108 + 3 - 12 = 99

c53 = a51 · b13 + a52 · b23 + a53 · b33 = 27 · 3 + 3 · 5 + 12 · (-4) = 81 + 15 - 48 = 48

c61 = a61 · b11 + a62 · b21 + a63 · b31 = 5 · 1 + 1 · (-2) + 18 · 2 = 5 - 2 + 36 = 39

c62 = a61 · b12 + a62 · b22 + a63 · b32 = 5 · 4 + 1 · 1 + 18 · (-1) = 20 + 1 - 18 = 3

c63 = a61 · b13 + a62 · b23 + a63 · b33 = 5 · 3 + 1 · 5 + 18 · (-4) = 15 + 5 - 72 = -52

c33 = a31 · b13 + a32 · b23 + a33 · b33 = 1 · 3 + 19 · 5 + 20 · (-4) = 3 + 95 - 80 = 18

c41 = a41 · b11 + a42 · b21 + a43 · b31 = 27 · 1 + 9 · (-2) + 19 · 2 = 27 - 18 + 38 = 47

c42 = a41 · b12 + a42 · b22 + a43 · b32 = 27 · 4 + 9 · 1 + 19 · (-1) = 108 + 9 - 19 = 98

c43 = a41 · b13 + a42 · b23 + a43 · b33 = 27 · 3 + 9 · 5 + 19 · (-4) = 81 + 45 - 76 = 50

c51 = a51 · b11 + a52 · b21 + a53 · b31 = 27 · 1 + 3 · (-2) + 12 · 2 = 27 - 6 + 24 = 45

c52 = a51 · b12 + a52 · b22 + a53 · b32 = 27 · 4 + 3 · 1 + 12 · (-1) = 108 + 3 - 12 = 99

c53 = a51 · b13 + a52 · b23 + a53 · b33 = 27 · 3 + 3 · 5 + 12 · (-4) = 81 + 15 - 48 = 48

c61 = a61 · b11 + a62 · b21 + a63 · b31 = 5 · 1 + 1 · (-2) + 18 · 2 = 5 - 2 + 36 = 39

c62 = a61 · b12 + a62 · b22 + a63 · b32 = 5 · 4 + 1 · 1 + 18 · (-1) = 20 + 1 - 18 = 3

c63 = a61 · b13 + a62 · b23 + a63 · b33 = 5 · 3 + 1 · 5 + 18 · (-4) = 15 + 5 - 72 = -52

So, the result of encoding is the number sequence 14 83 80 51 96 36 3 3 18 47 98 50 45 99 48 39 3 -52