Use the given coding matrices to encode and then decode the given message HELP. A=begin{bmatrix}4 & -1 -3 & 1 end{bmatrix} and its inverse A^{-1}=begin{bmatrix}1 & 1 3 & 4 end{bmatrix}

Use the given coding matrices to encode and then decode the given message HELP.
$A=\left[\begin{array}{cc}4& -1\\ -3& 1\end{array}\right]$ and its inverse ${A}^{-1}=\left[\begin{array}{cc}1& 1\\ 3& 4\end{array}\right]$
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Step 1
Concept:
A rectangular array of numbers (or other mathematical objects) for those the operations such as addition and multiplication are defined is called matrices. Matrix are used for plotting graph, statistics and also to do scientific studies and research in almost different fields.
Step 2
Given:
$A=\left[\begin{array}{cc}4& -1\\ -3& 1\end{array}\right]$ and its inverse
${A}^{-1}=\left[\begin{array}{cc}1& 1\\ 3& 4\end{array}\right]$
Step 3
Expressing the word numerically
The numerical equivalent of HELP is 8,5,12,16
Listing the numbers is the previous step by columns and form a square matrix
The $2×2$ matric formed by the numbers 8,5,12 and 16 is $\left[\begin{array}{cc}8& 12\\ 5& 16\end{array}\right]$
Step 4
Multiplying the matrix obtained by the given coding matrix A
$\left[\begin{array}{cc}4& -1\\ -3& 1\end{array}\right]\left[\begin{array}{cc}8& 12\\ 5& 16\end{array}\right]=\left[\begin{array}{cc}4\left(8\right)+\left(-1\right)\left(5\right)& 4\left(12\right)+\left(-1\right)\left(16\right)\\ \left(-3\right)\left(8\right)+1\left(5\right)& \left(-3\right)\left(12\right)+1\left(16\right)\end{array}\right]$
$=\left[\begin{array}{cc}27& 32\\ -19& -20\end{array}\right]$
Step 5
The encoded message is 27,-19,32,-20
The inverse is given as
${A}^{-1}=\left[\begin{array}{cc}1& 1\\ 3& 4\end{array}\right]$
$\left[\begin{array}{cc}1& 1\\ 3& 4\end{array}\right]\left[\begin{array}{cc}27& 32\\ -19& -20\end{array}\right]=\left[\begin{array}{cc}1\left(27\right)+1\left(-19\right)& 1\left(32\right)+1\left(-20\right)\\ 3\left(27\right)+4\left(-19\right)& 3\left(32\right)+4\left(-20\right)\end{array}\right]$
$=\left[\begin{array}{cc}8& 12\\ 5& 16\end{array}\right]$
Step 6
The numbers are 8,5,12 and 16
Therefore, the decoded message is HELP
Jeffrey Jordon