Question

Clasa a 11 a , va rog mult , dau coroana!

Answer 1

$\displaystyle \mathtt{A= \left(\begin{array}{ccc}\mathtt m&\mathtt2&\mathtt{-2}\\\mathtt3&\mathtt1&\mathtt{-1}\\\mathtt0&\mathtt2&\mathtt m\end{array}\right),~m\in\mathbb{R}}\\ \\ \mathtt{Matricea~A~este~inversabil\u{a}\Leftrightarrow det(A)\ne0}$

$\displaystyle \mathtt{det(A)=\left|\begin{array}{ccc}\mathtt m&\mathtt2&\mathtt{-2}\\\mathtt3&\mathtt1&\mathtt{-1}\\\mathtt0&\mathtt2&\mathtt m\end{array}\right|=m \cdot 1 \cdot m+(-2)\cdot3\cdot2+2\cdot(-1)\cdot0-}\\ \\ \mathtt{-(-2)\cdot1\cdot0-2\cdot3\cdot m-m\cdot(-1)\cdot2=m^2-12+0+0-6m+2m=}\\ \\ \mathtt{=m^2-4m-12}\\ \\ \mathtt{det(A)=m^2-4m-12}$

$\displaystyle \mathtt{det(A)\ne0\Rightarrow m^2-4m-12\ne0}\\ \\ \mathtt{m^2-4m-12=0}\\ \\ \mathtt{m^2-6m+2m-12=0}\\ \\ \mathtt{m(m-6)+2(m-6)=0}\\ \\ \mathtt{(m-6)(m+2)=0}\\ \\ \mathtt{m=6;~m=-2}\\ \\ \mathtt{det(A)\ne 0\Leftrightarrow m\in\mathbb{R}-\{-2;6\}}$