Question

Pentru fiecare numar real a se considera matricea M (a) = 2a O
O 2a
a) aratati ca M 1pe2 + M -1pe2 = M(0)

b) determinati numarul real a pt. care det (M(a)) =0

c) determinati matricea M(-2)+M(-1)+M(0)+M(1)+M(2)

Answer 1

$M(a)= \left(\begin{array}{ccc}2a&0\\0&2a\end{array}\right).$

a)

$M\left(\frac{1}{2}\right)+M\left(-\frac{1}{2}\right)= \left(\begin{array}{ccc}1&0\\0&1\end{array}\right) + \left(\begin{array}{ccc}-1&0\\0&-1\end{array}\right) =\\ \\ \\ =\left(\begin{array}{ccc}0&0\\0&0\end{array}\right) = M(0).$

b)

$\det M(a)=4a^2.\\ \\ 4a^2=0 \ \ \ \Rightarrow \ \ \ a=0.$

c)

 $M(-2)+M(-1)+M(0)+M(1)+M(2) = \\ \\ \\ = \left(\begin{array}{ccc}-4&0\\0&-4\end{array}\right) + \left(\begin{array}{ccc}-2&0\\0&-2\end{array}\right)+\\ \\ + \left(\begin{array}{ccc}-0&0\\0&-0\end{array}\right)+\\ \\ + \left(\begin{array}{ccc}2&0\\0&2\end{array}\right)+ \left(\begin{array}{ccc}4&0\\0&4\end{array}\right) = \\ \\ \\ = \left(\begin{array}{ccc}0&0\\0&0\end{array}\right).$