Question

3. Write a program in python that will determine the inverse mod 26 of $\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$. The naïve method for calculating multiplicative inverse of 'a' mod m is given below: def modInverse(a, m) : a = a % m; for x in range(1, m) : if ((a * x) % m == 1): return x return 1 Instead of this you may use Euclid's Extend algorithm to find modular inverse. Example: IfA = $\begin{bmatrix} 5 & 8 \\ 17 & 3 \end{bmatrix}$, then find A?¹mod 26. Now det(A)={(5 × 3) - (8 × 17)} mod 26 = -121 mod 26 = 9 Cofactor matrix of A = $\begin{bmatrix} 3 & -17 \\ -8 & 5 \end{bmatrix}$ and Adjoint of cofactor matrix is $\begin{bmatrix} 3 & -8 \\ -17 & 5 \end{bmatrix}$ So A?¹ = (1/det(A)) X Adj(cofactor(A))= (1/9) $\begin{bmatrix} 3 & -8 \\ -17 & 5 \end{bmatrix}$ mod 26 (1/9) mod 26=9?¹ mod 26=3 So A?¹=3$\begin{bmatrix} 3 & -8 \\ -17 & 5 \end{bmatrix}$ mod 26 = 3$\begin{bmatrix} 3 & 18 \\ 9 & 5 \end{bmatrix}$ mod 26=[$\begin{bmatrix} 9 & 54 \\ 27 & 15 \end{bmatrix}$ mod 26=$\begin{bmatrix} 9 & 2 \\ 1 & 15 \end{bmatrix}$

          3. Write a program in python that will determine the inverse mod 26 of $\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$.
The naïve method for calculating multiplicative inverse of 'a' mod m is given below:
def modInverse(a, m) :
    a = a % m;
    for x in range(1, m) :
        if ((a * x) % m == 1):
            return x
    return 1
Instead of this you may use Euclid's Extend algorithm to find modular inverse.
Example: IfA = $\begin{bmatrix} 5 & 8 \\ 17 & 3 \end{bmatrix}$, then find A?¹mod 26.
Now det(A)={(5 × 3) - (8 × 17)} mod 26 = -121 mod 26 = 9
Cofactor matrix of A = $\begin{bmatrix} 3 & -17 \\ -8 & 5 \end{bmatrix}$ and Adjoint of cofactor matrix is $\begin{bmatrix} 3 & -8 \\ -17 & 5 \end{bmatrix}$
So A?¹ = (1/det(A)) X Adj(cofactor(A))= (1/9) $\begin{bmatrix} 3 & -8 \\ -17 & 5 \end{bmatrix}$ mod 26
(1/9) mod 26=9?¹ mod 26=3
So A?¹=3$\begin{bmatrix} 3 & -8 \\ -17 & 5 \end{bmatrix}$ mod 26 = 3$\begin{bmatrix} 3 & 18 \\ 9 & 5 \end{bmatrix}$ mod 26=[$\begin{bmatrix} 9 & 54 \\ 27 & 15 \end{bmatrix}$ mod 26=$\begin{bmatrix} 9 & 2 \\ 1 & 15 \end{bmatrix}$

3. Write a program in python that will determine the inverse mod 26 of < p m a t r i x >.
The naïve method for calculating multiplicative inverse of 'a' mod m is given below:
def modInverse(a, m) :
a = a % m;
for x in range(1, m) :
if ((a * x) % m == 1):
return x
return 1
Instead of this you may use Euclid's Extend algorithm to find modular inverse.
Example: IfA = < b m a t r i x >, then find A?¹mod 26.
Now det(A)=(5 × 3) - (8 × 17) mod 26 = -121 mod 26 = 9
Cofactor matrix of A = < b m a t r i x > and Adjoint of cofactor matrix is < b m a t r i x >
So A?¹ = (1/det(A)) X Adj(cofactor(A))= (1/9) < b m a t r i x > mod 26
(1/9) mod 26=9?¹ mod 26=3
So A?¹=3< b m a t r i x > mod 26 = 3< b m a t r i x > mod 26=[< b m a t r i x > mod 26=< b m a t r i x >

Added by Zachary M.

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