Question

Determinați matricele A ∈ M2(IR) pentru care A + t(A) = O2.

Answer 1

$Fie,A= \left[\begin{array}{ccc}x&y\\z&t\\\end{array}\right]$, Matricea transpusa este: $A^{t}= \left[\begin{array}{ccc}x&z\\y&t\\\end{array}\right]$, Suma lor: $A+ A^{t}= \left[\begin{array}{ccc}2x&y+z&\\z+y&2t&\end{array}\right]=$$\left[\begin{array}{ccc}0&0\\0&0\end{array}\right]$. Rezulta, egalind elementele ce ocupa aceeas pozitie : 2x=2t=0 deci x=t=0, si z+y=y+z=0, adica y=-z, deci solutiile sunt matricele de forma : $A= \left[\begin{array}{ccc}0&a\\-a&0\end{array}\right]$, cu a∈R.