Solutions des exercices du TD 3

Exercice 1

1) $$ 2C - D = \begin{pmatrix} 6 & 6 & 13 \\ 11 & 1 & -7 \\ 5 & 4 & -2 \end{pmatrix}, \quad CD = \begin{pmatrix} 39 & 25 & 43 \\ 11 & 35 & -15 \\ 26 & 0 & 25 \end{pmatrix}, \quad C^2 = \begin{pmatrix} 74 & 39 & 24 \\ 24 & 41 & 48 \\ 44 & 11 & 18 \end{pmatrix} $$ $$ tA = \begin{pmatrix} -1 & 3 \\ 2 & 4 \end{pmatrix}, \quad tC = \begin{pmatrix} 4 & 8 & 3 \\ 5 & -1 & 4 \\ 6 & 0 & 0 \end{pmatrix}, \quad \text{Tr}(B) = -3, \quad \text{Tr}(D) = 1 $$ 2) $$ \det(A) = -10, \quad \det(D) = -103 $$ 3) $$ AB = \begin{pmatrix} 10 & 0 \\ 0 & 10 \end{pmatrix} = 10 I_2 \Rightarrow A^{-1} = \frac{1}{10} B = \begin{pmatrix} -\frac{2}{5} & \frac{1}{5} \\ \frac{3}{10} & \frac{1}{10} \end{pmatrix} $$

Exercice 2

$$ A^3 = 3A^2 - 5A + 2I \Rightarrow A(A^2 - 3A + 5I) = 2I \Rightarrow A^{-1} = \frac{1}{2}(A^2 - 3A + 5I) $$