# Find the inverse of the transformation x'= 2x - 3y, y' = x + y, that is, find x, y in terms of x' , y' . (Hint: Use matrices.) Is the transformation orthogonal?

Find the inverse of the transformation \(x=
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Aamina Herring
Step 1
To Determine:
Find the inverse of the transformation ${x}^{\prime }=2x-3y,{y}^{\prime }=x+y$, that is, find x, y in terms of x' , y' . (Hint: Use matrices.) Is the transformation orthogonal?
Given: we have ${x}^{\prime }=2x-3y$ and ${y}^{\prime }=x+y$
Explanation: we can write down the above x' and y' in matrix form
$\left(\begin{array}{c}{x}^{\prime }\\ {y}^{\prime }\end{array}\right)=\left(\begin{array}{cc}2& -3\\ 1& 1\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$
$\left(\begin{array}{c}x\\ y\end{array}\right)={\left(\begin{array}{cc}2& -3\\ 1& 1\end{array}\right)}^{-1}\left(\begin{array}{c}{x}^{\prime }\\ {y}^{\prime }\end{array}\right)$
Step 2
Determinate will be
$|\begin{array}{cc}2& -3\\ 1& 1\end{array}|=2-\left(-3\right)=5$
$\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{cc}\frac{1}{5}& \frac{3}{5}\\ -\frac{1}{5}& \frac{2}{5}\end{array}\right)\left(\begin{array}{c}{x}^{\prime }\\ {y}^{\prime }\end{array}\right)$
$\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{cc}\frac{{x}^{\prime }}{5}& \frac{3{y}^{\prime }}{5}\\ -\frac{{x}^{\prime }}{5}& \frac{2{y}^{\prime }}{5}\end{array}\right)$
Jeffrey Jordon