Exercice 3

1) $$ A = \begin{pmatrix} 2 & 0 & -2 \\ -4 & 6 & 8 \\ 0 & 2 & 2 \end{pmatrix}, \quad \det(A) = 8 $$ $$ A^{-1} = \frac{1}{8} \begin{pmatrix} -4 & -4 & 12 \\ 8 & 4 & -8 \\ -8 & -4 & 12 \end{pmatrix} = \begin{pmatrix} -\frac{1}{2} & -\frac{1}{2} & \frac{3}{2} \\ 1 & \frac{1}{2} & -1 \\ -1 & -\frac{1}{2} & \frac{3}{2} \end{pmatrix} $$ 2) \[ \begin{cases} 2x - 2z = 2 \\ -4x + 6y + 8z = 4 \\ 2y + 2z = 6 \end{cases} \Rightarrow A \cdot \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 2 \\ 4 \\ 6 \end{pmatrix} \] $$ \begin{pmatrix} x \\ y \\ z \end{pmatrix} = A^{-1} \cdot \begin{pmatrix} 2 \\ 4 \\ 6 \end{pmatrix} = \begin{pmatrix} 6 \\ -2 \\ 5 \end{pmatrix} $$

Exercice 4

1) \[ f : \mathbb{R}^2 \rightarrow \mathbb{R}^3, \quad f(x, y) = (3x - y,\ 2x + 4y,\ 5x - 6y) \] $$ M = \begin{pmatrix} 3 & -1 \\ 2 & 4 \\ 5 & -6 \end{pmatrix} $$ 2) \[ f : \mathbb{R}^3 \rightarrow \mathbb{R}^3, \quad f(x, y, z) = (3x - 2y + 5z,\ 2x + y - 7z,\ 4x - 6y) \] $$ M = \begin{pmatrix} 3 & -2 & 5 \\ 2 & 1 & -7 \\ 4 & -6 & 0 \end{pmatrix} $$