Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![3. Consider the linear transformations \( T : \mathbb{R}^2 \to \mathbb{R}^2 \) and \( S : \mathbb{R}^2 \to \mathbb{R}^2 \) defined by
\[
T \left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right) = \begin{bmatrix} 2x_2 \\ 3x_1 \end{bmatrix} \quad S \left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right) = \begin{bmatrix} x_2 \\ x_1 - 3x_2 \end{bmatrix} .
\]
Find \( T^{-1} \) and \( S^{-1} \). Using your answers, write a formula for \( (TS)^{-1} \).](https://content.bartleby.com/qna-images/question/67429e19-8d15-47bc-bbfe-dfa923849540/a89d97ab-7bce-4e7b-a568-ea3adf08c9ef/xj4og7b_processed.jpeg)
Transcribed Image Text:3. Consider the linear transformations \( T : \mathbb{R}^2 \to \mathbb{R}^2 \) and \( S : \mathbb{R}^2 \to \mathbb{R}^2 \) defined by
\[
T \left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right) = \begin{bmatrix} 2x_2 \\ 3x_1 \end{bmatrix} \quad S \left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right) = \begin{bmatrix} x_2 \\ x_1 - 3x_2 \end{bmatrix} .
\]
Find \( T^{-1} \) and \( S^{-1} \). Using your answers, write a formula for \( (TS)^{-1} \).
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