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For each set of conditions below, figure out whether there exist 3x3 matrices \(\mathbf{A}, \mathbf{B}, \mathbf{C}, \mathbf{D}\) satisfying them: \(\begin{align*} &\mathbf{A} \mathbf{v}  =  \mathbf{v} + \begin{pmatrix} 1\\ 2 \\ 3 \end{pmatrix} \text{ for all $\mathbf{v}$},\\ &\mathbf{B} \mathbf{v} = 2 \mathbf{v} \text{ for all $\mathbf{v}$},\\ &\mathbf{C} \begin{pmatrix}1 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix}1 \\ 0 \\ 0 \end{pmatrix}, \mathbf{C} \begin{pmatrix}0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix}0 \\ 1 \\ 0 \end{pmatrix},\mathbf{C} \begin{pmatrix}1 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix}1 \\ 1 \\ 0 \end{pmatrix},\\ &\mathbf{D} \begin{pmatrix}1 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix}1 \\ 2 \\ 3 \end{pmatrix}, \mathbf{D} \begin{pmatrix}0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix}2 \\ 3 \\ 4 \end{pmatrix} ,\mathbf{D} \begin{pmatrix}1 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 3\\ 4 \\ 5 \end{pmatrix}. \end{align*}\)

For each matrix, enter "yes" or "no" in the order above. (Enter "yes" if there is a matrix satisfying the given conditions.)

 

Could anyone help or give a pointer? Thank you so much ahh!

 Mar 2, 2020
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The answers are yes, yes, no, yes.

 Mar 2, 2020

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