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Sagot :
To find the secret phrase encoded by Amir, we need to decode the text using the provided matrices. Here is the step-by-step solution:
### Step 1: Write Down the Given Matrices
We have two matrices:
1. The encoding matrix [tex]\( C \)[/tex]:
[tex]\[ C = \begin{pmatrix} -3 & 2 \\ -4 & 3 \end{pmatrix} \][/tex]
2. The encoded text matrix:
[tex]\[ \begin{pmatrix} 10 & -26 & -49 & 9 & 3 & -13 & -50 \\ 16 & -33 & -63 & 16 & 9 & -10 & -65 \end{pmatrix} \][/tex]
### Step 2: Find the Inverse of the Encoding Matrix [tex]\( C \)[/tex]
To decode the matrix, we need to find the inverse of [tex]\( C \)[/tex]:
The inverse of a [tex]\( 2 \times 2 \)[/tex] matrix [tex]\(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\)[/tex] is given by:
[tex]\[ \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \][/tex]
Plugging in the values for [tex]\( C \)[/tex]:
[tex]\[ \text{Det}(C) = (-3)(3) - (2)(-4) = -9 + 8 = -1 \][/tex]
So, the inverse [tex]\( C^{-1} \)[/tex] is:
[tex]\[ C^{-1} = \frac{1}{-1} \begin{pmatrix} 3 & -2 \\ 4 & -3 \end{pmatrix} = \begin{pmatrix} -3 & 2 \\ -4 & 3 \end{pmatrix} \][/tex]
### Step 3: Multiply the Encoded Matrix by the Inverse of [tex]\( C \)[/tex]
Let [tex]\( E \)[/tex] be the encoded matrix:
[tex]\[ E = \begin{pmatrix} 10 & -26 & -49 & 9 & 3 & -13 & -50 \\ 16 & -33 & -63 & 16 & 9 & -10 & -65 \end{pmatrix} \][/tex]
We need to calculate [tex]\( C^{-1} \times E \)[/tex]:
[tex]\[ C^{-1} \times E = \begin{pmatrix} -3 & 2 \\ -4 & 3 \end{pmatrix} \times \begin{pmatrix} 10 & -26 & -49 & 9 & 3 & -13 & -50 \\ 16 & -33 & -63 & 16 & 9 & -10 & -65 \end{pmatrix} \][/tex]
### Step 4: Decode the Numerical Matrix into Letters
Once we obtain the resulting matrix after the multiplication, we map the numerical values back to letters according to [tex]\( A=1, B=2, C=3, \ldots, Z=26 \)[/tex], and assume [tex]\( 0 \)[/tex] represents a space.
### Step 5: Check with Possible Phrases
Given the possible phrases are:
1. THE BEAN IS GREEN
2. BLUE IS THE GLOVE
3. GREEN IS THE BEAN
4. THE GLOVE IS BLUE
The decoded numerical values will map to one of these phrases.
### Conclusion
After decoding the matrix and analyzing the numerical values, you will find that the correct secret phrase is:
[tex]\[ \boxed{\text{BLUE IS THE GLOVE}} \][/tex]
### Step 1: Write Down the Given Matrices
We have two matrices:
1. The encoding matrix [tex]\( C \)[/tex]:
[tex]\[ C = \begin{pmatrix} -3 & 2 \\ -4 & 3 \end{pmatrix} \][/tex]
2. The encoded text matrix:
[tex]\[ \begin{pmatrix} 10 & -26 & -49 & 9 & 3 & -13 & -50 \\ 16 & -33 & -63 & 16 & 9 & -10 & -65 \end{pmatrix} \][/tex]
### Step 2: Find the Inverse of the Encoding Matrix [tex]\( C \)[/tex]
To decode the matrix, we need to find the inverse of [tex]\( C \)[/tex]:
The inverse of a [tex]\( 2 \times 2 \)[/tex] matrix [tex]\(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\)[/tex] is given by:
[tex]\[ \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \][/tex]
Plugging in the values for [tex]\( C \)[/tex]:
[tex]\[ \text{Det}(C) = (-3)(3) - (2)(-4) = -9 + 8 = -1 \][/tex]
So, the inverse [tex]\( C^{-1} \)[/tex] is:
[tex]\[ C^{-1} = \frac{1}{-1} \begin{pmatrix} 3 & -2 \\ 4 & -3 \end{pmatrix} = \begin{pmatrix} -3 & 2 \\ -4 & 3 \end{pmatrix} \][/tex]
### Step 3: Multiply the Encoded Matrix by the Inverse of [tex]\( C \)[/tex]
Let [tex]\( E \)[/tex] be the encoded matrix:
[tex]\[ E = \begin{pmatrix} 10 & -26 & -49 & 9 & 3 & -13 & -50 \\ 16 & -33 & -63 & 16 & 9 & -10 & -65 \end{pmatrix} \][/tex]
We need to calculate [tex]\( C^{-1} \times E \)[/tex]:
[tex]\[ C^{-1} \times E = \begin{pmatrix} -3 & 2 \\ -4 & 3 \end{pmatrix} \times \begin{pmatrix} 10 & -26 & -49 & 9 & 3 & -13 & -50 \\ 16 & -33 & -63 & 16 & 9 & -10 & -65 \end{pmatrix} \][/tex]
### Step 4: Decode the Numerical Matrix into Letters
Once we obtain the resulting matrix after the multiplication, we map the numerical values back to letters according to [tex]\( A=1, B=2, C=3, \ldots, Z=26 \)[/tex], and assume [tex]\( 0 \)[/tex] represents a space.
### Step 5: Check with Possible Phrases
Given the possible phrases are:
1. THE BEAN IS GREEN
2. BLUE IS THE GLOVE
3. GREEN IS THE BEAN
4. THE GLOVE IS BLUE
The decoded numerical values will map to one of these phrases.
### Conclusion
After decoding the matrix and analyzing the numerical values, you will find that the correct secret phrase is:
[tex]\[ \boxed{\text{BLUE IS THE GLOVE}} \][/tex]
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