Question

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Answer 1

Răspuns:

Explicație pas cu pas:

a)

$det A=\left|\begin{array}{cc}2&4\\-1&-2\end{array}\right|=2*(-2)-4*(-1)=-4+4=0$

b)

$M(x)*M(1)=(A+xB)*(A+B)=A^2+AB+xBA+xB^2\\A^2=A*A=\left[\begin{array}{cc}2&4\\-1&-2\end{array}\right]*\left[\begin{array}{cc}2&4\\-1&-2\end{array}\right]=\left[\begin{array}{cc}2*2+4*(-1)&2*4+4*(-2)\\-1*2+(-1)*(-2)&-1*4+(-2)*(-2)\end{array}\right]=\left[\begin{array}{cc}0&0\\0&0\end{array}\right]$$AB=\left[\begin{array}{cc}2&4\\-1&-2\end{array}\right]*\left[\begin{array}{cc}-2&-6\\1&3\end{array}\right]=\left[\begin{array}{cc}2*(-2)+4*1&2*(-6)+4*3\\-1*(-2)+(-2)*1&-1*(-6)+(-2)*3\end{array}\right]=\left[\begin{array}{cc}0&0\\0&0\end{array}\right]$$BA=\left[\begin{array}{cc}-2&-6\\1&3\end{array}\right]*\left[\begin{array}{cc}2&4\\-1&-2\end{array}\right]=\left[\begin{array}{cc}-2*2+(-6)*(-1)&-2*4+(-6)*(-2)\\1*2+3*(-1)&1*4+3*(-2)\end{array}\right]=\left[\begin{array}{cc}2&4\\-1&-2\end{array}\right]=A$$B^2=B*B=\left[\begin{array}{cc}-2&-6\\1&3\end{array}\right]*\left[\begin{array}{cc}-2&-6\\1&3\end{array}\right]=\left[\begin{array}{cc}-2*(-2)+(-6)*1&-2*(-6)+3*(-6)\\1*(-2)+3*1&1*(-6)+3*3\end{array}\right]=\left[\begin{array}{cc}-2&-6\\1&3\end{array}\right]=B$$M(x)*M(1)=O_2+O_2+xA+xB=x(A+B)=x*M(1)$

c)

M(4)*M(3)*M(2)*M(1)=nM(1) 1)

Conform punctului b) M(x)*M(1)=xM(1),∀ x∈R => M(4)*M(3)*M(2)*M(1)=M(4)*M(3)*2*M(1)=2M(4)*M(3)*M(1)=2*M(4)*3*M(1)=2*3*M(4)*M(1)=2*3*4*M(1)=24M(1)   2)

Din 1) si 2) => 24M(1)=nM(1)=> n=24