Se consideră matricele [tex]$A=\left(\begin{array}{rr}4 & -6 \\ 2 & -3\end{array}\right), I_{2}=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$[/tex] şi [tex]$M(x)=I_{2}+x A$[/tex], unde [tex]$x$[/tex] este număr real.

[tex]$5 \mathbf{a}$[/tex] a) Arătați că det [tex]$A=0$[/tex].

[tex]$5 \mathbf{p}$[/tex] b) Demonstrați că [tex]$M(x) M(y)=M(x+y+x y)$[/tex], pentru orice numere reale [tex]$x$[/tex] și [tex]$y$[/tex].

[tex]$5 \mathbf{p}$[/tex] c) Determinați perechile de numere naturale [tex]$(m, n)$[/tex] pentru care [tex]$M(m) M(n)=M(6)$[/tex].

[tex]M(x)=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)+x\left(\begin{array}{rr}4 & -6 \\ 2 & -3\end{array}\right)=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)+\left(\begin{array}{rr}4x & -6x \\ 2x & -3x\end{array}\right)\\\\M(x)=\left(\begin{array}{rr}1+4x & -6x \\ 2x & 1-3x\end{array}\right)[/tex]

[tex]M(y)=\left(\begin{array}{rr}1+4y & -6y \\ 2y & 1-3y\end{array}\right)[/tex]

[tex]M(x)M(y)=\left(\begin{array}{rr}1+4x & -6x \\ 2x & 1-3x\end{array}\right)\times \left(\begin{array}{rr}1+4y & -6y \\ 2y & 1-3y\end{array}\right)=\\\\=\left(\begin{array}{rr}(1+4x)(1+4y)-12xy & -6y-24xy-6x+18xy \\ 2x+8xy+2y-6xy & -12xy+(1-3x)(1-3y)\end{array}\right)[/tex]

[tex]M(x)M(y)=\left(\begin{array}{rr}1+4x+4y+4xy & -6x-6y-6xy \\ 2x+2y+2xy & 1-3x-3y-3xy\end{array}\right)\\\\M(x)M(y)=\left(\begin{array}{rr}1+4(x+y+xy) & -6(x+y+xy) \\ 2(x+y+xy) & 1-3(x+y+xy)\end{array}\right)=M(x+y+xy)[/tex]

c)

Ne folosim de punctul b, stim ca M(x)M(y)=M(x+y+xy)

M(m)M(n)=M(m+n+mn)

M(m+n+mn)=M(6)

m+n+mn=6

m+n+mn+1-1=6

m+1+n(m+1)=7

(m+1)(n+1)=7

I.

m+1=7

m=6

n+1=1

n=0

II.

m+1=1

m=0

n+1=7

n=6

Un exercitiu similar de bac gasesti aici: https://brainly.ro/tema/2494494

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