+0

# For each set of conditions below, figure out whether there exist 3x3 matrices satisfying them:

-1
39
1
+82

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