Question

1. Se consideră matricea A = ( 2    -2) { pe primul rand } 
                                              1   -1   { pe al doilea rand }
a. Calculati Det A
b. determinati numerele reale p pentru A x A = pA
c. Det matricile B = ( 0    b ) 
                               b    0 ) stiind ca det ( A + B) = 0 

Answer 1

[tex] A=\left[\begin{array}{cc}2&-2\\1&-1\end{array}\right] \\ \det(A)=2\cdot (-1) -(-2)\cdot1=0\\ A^2=\left[\begin{array}{cc}2&-2\\1&-1\end{array}\right]\left[\begin{array}{cc}2&-2\\1&-1\end{array}\right]=\left[\begin{array}{cc}2&-2\\1&-1\end{array}\right].\\ \text{de unde } p=1.\\ A+B=\left[\begin{array}{cc}2&-2\\1&-1\end{array}\right]+\left[\begin{array}{cc}0&b\\b&0\end{array}\right]=\left[\begin{array}{cc}2&b-2\\1+b&-1\end{array}\right]\\ \det(A+B)=2\cdot(-1)-(b-2)(1+b)=0\\ \Rightarrow b=0 \vee b=1[/tex]