To determine by which matrix Dina should multiply the encoded text to retrieve the clear text code, we need to find the inverse of the encoding matrix [tex]\( C \)[/tex].
Given the encoding matrix [tex]\( C \)[/tex] is:
[tex]\[ C = \begin{bmatrix}
-2 & 1 \\
3 & -1
\end{bmatrix} \][/tex]
The inverse of a 2x2 matrix [tex]\( C \)[/tex], where
[tex]\[ C = \begin{bmatrix}
a & b \\
c & d
\end{bmatrix} \][/tex]
is calculated using the formula:
[tex]\[ C^{-1} = \frac{1}{ad - bc} \begin{bmatrix}
d & -b \\
-c & a
\end{bmatrix} \][/tex]
Plugging in our values where [tex]\( a = -2 \)[/tex], [tex]\( b = 1 \)[/tex], [tex]\( c = 3 \)[/tex], and [tex]\( d = -1 \)[/tex], we compute:
1. Calculate the determinant [tex]\( ad - bc \)[/tex]:
[tex]\[ \text{Det}(C) = (-2 \cdot -1) - (1 \cdot 3) = 2 - 3 = -1 \][/tex]
2. Substitute into the inverse matrix formula:
[tex]\[ C^{-1} = \frac{1}{-1} \begin{bmatrix}
-1 & -1 \\
-3 & -2
\end{bmatrix}
= \begin{bmatrix}
1 & 1 \\
3 & 2
\end{bmatrix} \][/tex]
Thus, the matrix that will convert the encoded text back to the clear text code is:
[tex]\[ \begin{bmatrix}
1 & 1 \\
3 & 2
\end{bmatrix} \][/tex]
So, the correct answer is:
[tex]\[ \left(\begin{array}{ll}1 & 1 \\ 3 & 2\end{array}\right] \][/tex]
