Question

$x^{2} =\left[\begin{array}{ccc}7&-5\\-15&12 \end{array}\right]$
Aflati matricea X

Answer 1

[tex] fie X= \left[\begin{array}{ccc}a&b\\c&d\end{array}\right] ; X^{2} = \left[\begin{array}{ccc}a&b\\c&d\end{array}\right] \left[\begin{array}{ccc}a&b\\c&d\end{array}\right] ; [/tex]
[tex]X^2= \left[\begin{array}{ccc}a^2+bc&ab+bd\\ac+cd&bc+d^2\end{array}\right] X^2= \left[\begin{array}{ccc}7&-5\\-15&12\end{array}\right] [/tex]
Din astea rezulta ca:[tex]a^2+bc=7; ab+bd=-5; ac+cd=-15; bc+d^2=12;[/tex]
Rezolvi sistemul si gasesti solutiile:
Solutia 1:a=-2; b=1; c=3; d=-3; 
Solutia 2:a=2; b=-1; c=-1; d=3;
Adica :
$X= \left[\begin{array}{ccc}-2&1\\3&-3\end{array}\right] sau \left[\begin{array}{ccc}2&-1\\-3&3\end{array}\right]$