Răspuns:
Explicație pas cu pas:
a) [tex]\left|\begin{array}{ccc}3&1\\9&3\end{array}\right | = 3*3-1*9=9-9=0[/tex]
b) X(a,b) = [tex]\left[\begin{array}{ccc}a&b&\\9b&a\end{array}\right][/tex]
X(c,d) = [tex]\left[\begin{array}{ccc}c&d&\\9d&c\end{array}\right][/tex]
X(a,b)*X(c,d) = [tex]\left[\begin{array}{ccc}a&b&\\9b&a\end{array}\right] * \left[\begin{array}{ccc}c&d&\\9d&c\end{array}\right] = \left[\begin{array}{ccc}ac+9bd&ad+bc&\\9bc+9ad&9bd+ac\end{array}\right] = \left[\begin{array}{ccc}ac+9bd&ad+bc&\\9(ad+bc)&ac+9bd\end{array}\right]=X(ac+9bd,ad+bc)[/tex]
c) |X(m,n)| = [tex]\left|\begin{array}{ccc}m&n&\\9n&m\end{array}\right| = 1[/tex] <=> m^{2}-9n^{2}=1 <=> (m-3n)(m+3n)=1 <=> m-3n=m+3n=1 sau m-3n=m+3n= -1
m=1+3n => 1+3n+3n=1 =>6n=0 => n=0 => m=1
sau m=3n-1 => 3n-1+3n=-1 => 6n=0 => n=0 => m= -1
Deci n=0 si m= ±1
