Question

A= $\left[\begin{array}{ccc}-1&1\\0&0\\\end{array}\right]$

Determinați numerele reale m pentru care detB = 0 , unde B = A·(A + m I₂)

Answer 1

$B=A\cdot(A+m\cdot I_2)=\left(\begin{array}{cc}-1&1\\0&0\end{array}\right)\cdot\left[\left(\begin{array}{cc}-1&1\\0&0\end{array}\right)+m\cdot\left(\begin{array}{cc}1&0\\0&1\end{array}\right)\right]=\\\\=\left(\begin{array}{cc}-1&1\\0&0\end{array}\right)\cdot\left[\left(\begin{array}{cc}-1&1\\0&0\end{array}\right)+\left(\begin{array}{cc}m&0\\0&m\end{array}\right)\right]=\left(\begin{array}{cc}-1&1\\0&0\end{array}\right)\cdot\left(\begin{array}{cc}m-1&1\\0&m\end{array}\right)=\\\\=\left(\begin{array}{cc}1-m+1\cdot 0&(-1)\cdot1+1\cdot m\\0\cdot(m-1)+0\cdot 0&0\cdot1+0\cdot m\end{array}\right)=\left(\begin{array}{cc}1-m&m-1\\0&0\end{array}\right).$

Din cele de mai sus detB = 0, deci pentru orice m ∈ R detB=0.

Green eyes.