Question

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Answer 1

$A+I=\left[\begin{array}{ccc}a+1&-b&-a\\0&2&2\\1&1&b+1\end{array}\right] \\ \\\text{Entonces }\\\det(A+I)= (-1)^{1+1}(a+1) (2b)+(-1)^{2+1}(0)(-b^2-b+a)+(-1)^{3+1}(1)(-2b+2a)\\\det(A+I)=(2ab+2b)+(-2b+2a)=2ab+2a=12\\\\ab+a=6\\\\a(b+1)=6\to (a,b)\in\{(1,5),(2,2),(3,1)\}\cap \{(x,y):x=2k~\&~ y=2k\}=\{(2,2)\}\\\\\det(M)=ab-2ab^2=2(2)-2(2)(4)\\\\\boxed{\det(M)=-12}$