Răspuns:
a)
[tex]\det(A^t)\det A=\begin{vmatrix}1 & 1 & 0\\0 & 0 & 1\\0 & 1 & 0\end{vmatrix}=\begin{vmatrix}0 & 1\\1 & 0\end{vmatrix}=-1[/tex]
b)
[tex]A^2=\begin{pmatrix}1 & 1 & 1\\0 & 1 & 0\\0 & 0 & 1\end{pmatrix}, \ A^3=A^2\cdot A=\begin{pmatrix}1 & 2 & 1\\0 & 0 & 1\\0 & 1 & 0\end{pmatrix}[/tex]
c)
[tex]A^2+A=\begin{pmatrix}1 & 1 & 1\\0 & 1 & 0\\0 & 0 & 1\end{pmatrix}+\begin{pmatrix}1 & 1 & 0\\0 & 0 & 1\\0 & 1 & 0\end{pmatrix}=\begin{pmatrix}2 & 2 & 1\\0 & 1 & 1\\0 & 1 & 1\end{pmatrix}[/tex]
[tex]A^3+I_3=\begin{pmatrix}2 & 2 & 1\\0 & 1 & 1\\0 & 1 & 1\end{pmatrix}+\begin{pmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{pmatrix}=\begin{pmatrix}2 & 2 & 1\\0 & 1 & 1\\0 & 1 & 1\end{pmatrix}[/tex]
Se observă că rezultatele sunt egale.
P.S. Păstrează-ți coroana!
Explicație pas cu pas:
