math

There is no matrix A that satisfies the 3 equations.

(1,1,1) = (1,0,0)+(0,1,1)

A(1,1,1) = A((1,0,0)+(0,1,1)) = A(1,0,0) + A(0,1,1) = (-3,4,0) + (1,2,3) = (-2, 6, 3) != (3,2,1)

1.

If there exists a matrix $\mathbf{A}$ such that
$\mathbf{A} \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} -3 \\ 4 \\ 0 \end{pmatrix},\ \mathbf{A} \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix},\ \mathbf{A} \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 3 \\ 2 \\ 1 \end{pmatrix}$
calculate
$\mathbf{A} \begin{pmatrix} 1 \\ -1 \\ -1 \end{pmatrix}$.

$\begin{array}{|c|c|c|c|} \hline & I. & II. & III. \\ \hline \mathbf{A} & \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} & \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} & \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \\ \hline \begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33} \end{pmatrix} & \begin{pmatrix} -3 \\ 4 \\ 0 \end{pmatrix} & \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} & \begin{pmatrix} 3 \\ 2 \\ 1 \end{pmatrix} \\ \hline \end{array}$

$\begin{array}{|l|l|l|} \hline I. & II.& III. \\ \hline a_{11} = -3 & \mathbf{a_{12}+a_{12} = \color{red}1} & \underbrace{a_{11}}_{-3}+a_{12}+a_{13} = 3 \qquad \mathbf{a_{12}+a_{13} = \color{red}6}\quad \text{contradiction!} \\ a_{21} = 4 & a_{22}+a_{23} = 2 & a_{21}+a_{22}+a_{23} = 2 \\ a_{31} = 0 & a_{32}+a_{33} = 3 & a_{31}+a_{32}+a_{33} = 1 \\ \hline \end{array}$

There's no such matrix $\mathbf{A}$

2.

$\mathbf{A} \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix},\ \mathbf{A} \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} -3 \\ 4 \\ 0 \end{pmatrix},\ \mathbf{A} \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} -2 \\ 6 \\ 3 \end{pmatrix}$

calculate
$\mathbf{A} \begin{pmatrix} 1 \\ -1 \\ -1 \end{pmatrix}$.

$\begin{array}{|c|c|c|c|} \hline & I. & II. & III. & IV. \\ \hline \mathbf{A} & \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} & \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} & \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} & \begin{pmatrix} 1 \\ -1 \\ -1 \end{pmatrix} \\ \hline \begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33} \end{pmatrix} & \begin{pmatrix} -3 \\ 4 \\ 0 \end{pmatrix} & \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} & \begin{pmatrix} -2 \\ 6 \\ 3 \end{pmatrix} \\ \hline \end{array}$

$\begin{array}{|l|l|lll|} \hline I. & II.& III. \\ \hline a_{11} = -3 & a_{12}+a_{12} = 1 & \underbrace{a_{11}}_{-3}+a_{12}+a_{13} = -2 & -3+ a_{12}+a_{13} = -2 & a_{12}+a_{13}=1\ \checkmark \\ a_{21} = 4 & a_{22}+a_{23} = 2 & \underbrace{a_{21}}_{4}+a_{22}+a_{23} = 6 & 4+ a_{22}+a_{23} = 6 & a_{22}+a_{23}=2\ \checkmark \\ a_{31} = 0 & a_{32}+a_{33} = 3 & \underbrace{a_{31}}_{0}+a_{32}+a_{33} = 3 & 0+ a_{32}+a_{33} = 3 & a_{32}+a_{33}=3\ \checkmark \\ \hline \end{array}$

$\begin{array}{|l|l|lll|} \hline IV. \\ \hline a_{11}-a_{21}-a_{13} = a_{11}-(a_{21}+a_{13}) = -3-1 = -4 \\ a_{21}-a_{22}-a_{23} = a_{21}-(a_{22}+a_{23}) = 4-2 = 2 \\ a_{31}-a_{32}-a_{33} = a_{31}-(a_{32}+a_{33}) = 0-3=-3 \\ \hline \end{array}$

$\mathbf{A} \begin{pmatrix} 1 \\ -1 \\ -1 \end{pmatrix} = \begin{pmatrix} -4 \\ 2 \\ -3 \end{pmatrix}$

Question 2.