Question

Determinați nr reale m știind ca det (A(m))=0
A(a)= 1 1 1
1 a 2
1 2 a

Answer 1

$\displaystyle \mathtt{A(a)= \left(\begin{array}{ccc}\mathtt1&\mathtt1&\mathtt1\\\mathtt1&\mathtt a&\mathtt2\\\mathtt1&\mathtt2&\mathtt a\end{array}\right)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~det(A(m))=0}$

$\displaystyle \mathtt{det(A(m))= \left|\begin{array}{ccc}\mathtt1&\mathtt1&\mathtt1\\\mathtt1&\mathtt m&\mathtt2\\\mathtt1&\mathtt2&\mathtt m\end{array}\right|=1 \cdot m \cdot m+1 \cdot 1 \cdot 2+1 \cdot2 \cdot 1-1 \cdot m \cdot 1-}\\ \\ \mathtt{-1 \cdot 1 \cdot m-1 \cdot 2 \cdot 2=m^2+2+2-m-m-4=m^2-2m}\\ \\ \mathtt{m^2-2m=0 \Rightarrow m(m-2)=0 \Rightarrow \mathbf{\underline{m=0}}}\\ \\ \mathtt{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\Rightarrow \mathbf{\underline{m=2}}}$