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Expert-verified Found in: Page 57 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # Question 8: Find the inverse of linear transformation ${{\mathbit{y}}}_{{\mathbf{1}}}{\mathbf{=}}{{\mathbit{x}}}_{{\mathbf{1}}}{\mathbf{+}}{\mathbf{7}}{{\mathbit{x}}}_{{\mathbf{2}}}\phantom{\rule{0ex}{0ex}}{{\mathbit{y}}}_{{\mathbf{2}}}{\mathbf{=}}{\mathbf{3}}{{\mathbit{x}}}_{{\mathbf{1}}}{\mathbf{+}}{\mathbf{20}}{{\mathbit{x}}}_{{\mathbf{2}}}$

Inverse of the transformation is${T}^{-1}\left({x}_{1},{x}_{2}\right)=\left(-20{x}_{1}+7{x}_{2},3{x}_{1}-{x}_{2}\right)$ .

See the step by step solution

## Step1: System of equations

We have given a linear transformation with

${y}_{1}={x}_{1}+7{x}_{2}\phantom{\rule{0ex}{0ex}}{y}_{2}=3{x}_{1}+20{x}_{2}$

## Step2: Linear Transformation

Let T be linear transformation from ${R}^{2}\to {R}^{2}$.

Then, according to given equations.

$T\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]=\left[\begin{array}{c}{x}_{1}+7{x}_{2}\\ 3{x}_{1}+20{x}_{2}\end{array}\right]$

## Step3: Matrix for linear transformation

The matrix representation of T by 2x2 matrix.

.$A=\left[\begin{array}{cc}1& 7\\ 3& 20\end{array}\right]$

Inverse of matrix is calculated as

First of all we will find the determinant of matrix.

$\left|A\right|=20-21=-1\phantom{\rule{0ex}{0ex}}adjA=\left[\begin{array}{cc}20& -7\\ -3& 1\end{array}\right]$

Inverse of matrix= .${A}^{-1}=\left[\begin{array}{cc}-20& 7\\ 3& -1\end{array}\right]$

Hence, inverse of the transformation will be

${T}^{-1}\left({x}_{1},{x}_{2}\right)=\left(-20{x}_{1}+7{x}_{2},3{x}_{1}-{x}_{2}\right)$

Hence, inverse of the transformation isdata-custom-editor="chemistry" ${T}^{-1}\left({x}_{1},{x}_{2}\right)=\left(-20{x}_{1}+7{x}_{2},3{x}_{1}-{x}_{2}\right)$ ### Want to see more solutions like these? 