\[\begin{aligned} & \operatorname{det} M_a=\left|\begin{array}{ccc} a+1 & -1 & -a \\ 1 & a & -1 \\ -1 & 0 & 2 a+1 \end{array}\right|=\left|\begin{array}{ccc} c_1+c_2+c_3 & -1 & -c_1 \\ a & a & -1 \\ 2 a & 0 & 2 a+1 \end{array}\right| \\ & =\underset{\substack{\text { mis en } \\ \text { facteur } \\ \text { de } c_2}}{ }\left|\begin{array}{ccc} 0 & -1 & -a \\ 1 & a & -1 \\ 2 & 0 & 2 a+1 \end{array}\right| \\ & =a .\left|\begin{array}{ccc} 0 & -1 & -a \\ 1 & a & -1 \\ 0 & -2 a & 2 a+3 \end{array}\right| \begin{array}{c} \ell_1 \\ \ell_3-2 \ell_2 \end{array} \\ & =a \cdot[-1 \times[(-1) \times(2 a+3)-(-2 a) \times(-a)]] \\ & =a \cdot \underbrace{\left(2 a^2+2 a+3\right)}_{\text {discriminant } \Delta>0} \\ & \end{aligned}\] \(M_a\) inversible ssi \(a \neq 0\). Calcul des cofacteurs : \[\begin{aligned} & M_{11}=\left|\begin{array}{cc} a & -1 \\ 0 & 2 a+1 \end{array}\right|=2 a^2+a \quad M_{12}=-\left|\begin{array}{cc} 1 & -1 \\ -1 & 2 a+1 \end{array}\right|=-2 a \quad M_{13}=\left|\begin{array}{cc} 1 & a \\ -1 & 0 \end{array}\right|=a \\ & M_{21}=-\left|\begin{array}{cc} -1 & -a \\ 0 & 2 a+1 \end{array}\right|=2 a+1 \quad M_{22}=\left|\begin{array}{cc} a+1 & -a \\ -1 & 2 a+1 \end{array}\right|=2 a^2+2 a+1 \quad M_{23}=-\left|\begin{array}{cc} a+1 & -1 \\ -1 & 0 \end{array}\right|=1 \\ & M_{31}=\left|\begin{array}{cc} -1 & -a \\ a & -1 \end{array}\right|=a^2+1 \quad M_{32}=-\left|\begin{array}{cc} a+1 & -a \\ 1 & -1 \end{array}\right|=1 \quad M_{33}=\left|\begin{array}{cc} a+1 & -1 \\ 1 & a \end{array}\right|=a^2+a+1 \\ & \end{aligned}\] on en déduit : \[\begin{aligned} \tilde{M} & =\left(\begin{array}{ccc} 2 a^2+a & -2 a & a \\ 2 a+1 & 2 a^2+2 a+1 & 1 \\ a^2+1 & 1 & a^2+a+1 \end{array}\right) \\ { }^t \tilde{M} & =\left(\begin{array}{ccc} 2 a^2+a & 2 a+1 & a^2+1 \\ -2 a & 2 a^2+2 a+1 & 1 \\ a & 1 & a^2+a+1 \end{array}\right) \end{aligned}\]

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Pour \(a \neq 0\) : \[M_a^{-1}=\frac{1}{a \cdot\left(2 a^2+2 a+3\right)}\left(\begin{array}{ccc} 2 a^2+a & 2 a+1 & a^2+1 \\ -2 a & 2 a^2+2 a+1 & 1 \\ a & 1 & a^2+a+1 \end{array}\right)\]

Matrices inversibles