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Part 1) Let \({A} = \begin{pmatrix} 3 & -2 & 3 \\ 1 & 2 & 1 \\ 1 & 3 & 0 \end{pmatrix},\)

and let \({v}_1 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \quad {v}_2 = \begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}, \quad {v}_3 = \begin{pmatrix} 11 \\ 1 \\ -14 \end{pmatrix}.\)

Show that A sends each of \(\mathbf{v}_1, \mathbf{v}_2, \) and \( \mathbf{v}_3\) to scalar multiples of themselves, and find the value of these scalars.

Part 2) Let n be a positive integer. Use part 1 to find the vectors \(\mathbf{A}^n \mathbf{v}_1, \mathbf{A}^n \mathbf{v}_2, \mathbf{A}^n\mathbf{v}_3.\)

 

Part 3) Calculate \({A}^{10} \begin{pmatrix} 10 \\ 4 \\ -11 \end{pmatrix}.\)

 Mar 16, 2020
edited by Guest  Mar 16, 2020
 #1
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Here's Part 1)

 

 Mar 16, 2020
 #2
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Here are parts 2 and 3:

 

 Mar 16, 2020
 #3
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Thank you!

Guest Mar 16, 2020

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