$($d$)$ By using the English alphabet $)=0,B=1,$dots,$Z=(25.$ Encrypt the message "ACT NOW" using the Hill cipher with the key $($K$).$

$K=\left[\begin{array}{ccc}6& 24& 1\\ 13& 16& 10\\ 20& 17& 15\end{array}\right]$

Decrypt the ciphertext, by first finding $K^{-1}$ and then apply the Hill cipher to find the plaintext. $($Note: Show your steps of calculation$).$

$($e$)$ Consider a modified Vigen$$re cipher where the set of characters are the English alphabet plus the numbers from $0$ to $9.$

$($i$)$ If $P_i$ is the $i^{th}$ digit of the plaintext, $C_i$ is the $i^{th}$ digit of the ciphertext, and $K_i$ is the $i^{th}$ digit of the key, write equations for the encryption and decryption operations.

$($ii$)$ For $P=$ BN$3$LS$68$MSZ$9$WE$2$ and keyword $=J7K5R,$ what is C$?$