Question

E is the encoding matrix.Use E to decode the coded message matrix.

Answer 1

To decode a message, we take the string of coded numbers and multiply it by the inverse of the matrix to get the original string of numbers. Finally, by associating the numbers with their corresponding letters, we obtain the original message.

The encoding matrix is given to be:

`E=\begin{bmatrix}{4} & {3} \\ {5} & {4}\end{bmatrix}`

The inverse of the matrix is calculated to be:

`E^(-1)=\begin{bmatrix}{4} & {-3} \\ {-5} & {4}\end{bmatrix}`

Next, we break the encoded matrix into single rows of 2 x 1 matrices and multiply each one with the inverse of the encoding matrix:

Row 1

`\Rightarrow\begin{bmatrix}{4} & {-3} \\ {-5} & {4}\end{bmatrix}\begin{bmatrix}{59} \\ {76}\end{bmatrix}=\begin{bmatrix}{8} \\ {9}\end{bmatrix}`

Row 2

`\Rightarrow\begin{bmatrix}{4} & {-3} \\ {-5} & {4}\end{bmatrix}\begin{bmatrix}{13} \\ {17}\end{bmatrix}=\begin{bmatrix}{1} \\ {3}\end{bmatrix}`

Row 3

`\Rightarrow\begin{bmatrix}{4} & {-3} \\ {-5} & {4}\end{bmatrix}\begin{bmatrix}{103} \\ {130}\end{bmatrix}=\begin{bmatrix}{22} \\ {5}\end{bmatrix}`

Row 4

`\Rightarrow\begin{bmatrix}{4} & {-3} \\ {-5} & {4}\end{bmatrix}\begin{bmatrix}{20} \\ {25}\end{bmatrix}=\begin{bmatrix}{5} \\ {0}\end{bmatrix}`

Row 5

`\Rightarrow\begin{bmatrix}{4} & {-3} \\ {-5} & {4}\end{bmatrix}\begin{bmatrix}{12} \\ {16}\end{bmatrix}=\begin{bmatrix}{0} \\ {4}\end{bmatrix}`

Row 6

`\Rightarrow\begin{bmatrix}{4} & {-3} \\ {-5} & {4}\end{bmatrix}\begin{bmatrix}{7} \\ {9}\end{bmatrix}=\begin{bmatrix}{1} \\ {1}\end{bmatrix}`

Row 7

`\Rightarrow\begin{bmatrix}{4} & {-3} \\ {-5} & {4}\end{bmatrix}\begin{bmatrix}{75} \\ {100}\end{bmatrix}=\begin{bmatrix}{0} \\ {25}\end{bmatrix}`

Row 8

`\Rightarrow\begin{bmatrix}{4} & {-3} \\ {-5} & {4}\end{bmatrix}\begin{bmatrix}{56} \\ {70}\end{bmatrix}=\begin{bmatrix}{14} \\ {0}\end{bmatrix}`

Therefore, the combined multiplied matrix is:

`\Rightarrow\begin{bmatrix}{8} \\ {9}\end{bmatrix}\begin{bmatrix}{1} \\ {3}\end{bmatrix}\begin{bmatrix}{22} \\ {5}\end{bmatrix}\begin{bmatrix}{5} \\ {0}\end{bmatrix}\begin{bmatrix}{0} \\ {4}\end{bmatrix}\begin{bmatrix}{1} \\ {1}\end{bmatrix}\begin{bmatrix}{0} \\ {25}\end{bmatrix}\begin{bmatrix}{14} \\ {0}\end{bmatrix}`

Comparing these with the letter codes shown below:

We have the code to be:

`\Rightarrow\begin{bmatrix}{H} \\ {I}\end{bmatrix}\begin{bmatrix}{A} \\ {C}\end{bmatrix}\begin{bmatrix}{V} \\ {E}\end{bmatrix}\begin{bmatrix}{E} \\ {-}\end{bmatrix}\begin{bmatrix}{-} \\ {D}\end{bmatrix}\begin{bmatrix}{A} \\ {A}\end{bmatrix}\begin{bmatrix}{-} \\ {Y}\end{bmatrix}\begin{bmatrix}{N} \\ {-}\end{bmatrix}`

The code is "HAVE A NICE DAY".

The THIRD OPTION is correct.