Matematică
dosied15
2015-11-04 02:36:24
Daca AB=BA,aratati că det B mai mic sau egal cu 0. Matricele A= (1 2 4 1) B=(0 x y 0)
Răspunsuri la întrebare
mariusbotezatu
2015-11-04 02:59:41

[latex]displaystyle A= left(egin{array}{ccc}1&2\4&1\ end{array} ight) \ \ B= left(egin{array}{ccc}0&x\y&0\ end{array} ight) \ \ AB=BA [/latex] [latex]displaystyle AB=left(egin{array}{ccc}1&2\4&1\ end{array} ight) cdot left(egin{array}{ccc}0&x\y&0\ end{array} ight) = left(egin{array}{ccc}1 cdot 0+2 cdot y&1 cdot x+2 cdot 0\4 cdot 0+1 cdot y&4cdot x+1 cdot 0 \ end{array} ight)= \ \ =left(egin{array}{ccc}2y&x\y&4x\end{array} ight) [/latex] [latex]BA=left(egin{array}{ccc}0&x\y&0\ end{array} ight) cdot left(egin{array}{ccc}1&2\4&1\ end{array} ight)=left(egin{array}{ccc}0 cdot 1+x cdot 4 &0 cdot 2 +x cdot 1\y cdot 1+0 cdot 4&y cdot 2+0cdot1\end{array} ight) = \ \ =left(egin{array}{ccc}4x&x\y&2y\end{array} ight) [/latex] [latex]2y=4x Rightarrow y=2x \ \ det~B= left|egin{array}{ccc}0&x\2x&0\end{array} ight|=0 cdot 0-x cdot 2x=0-2x^2=-2x^2 leq 0[/latex]

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