Question

Help! Punctul c, va rog :)

Answer 1

Calculam det(A(m)).
$det(A(m))=$$\left|\begin{array}{ccc}m-1&-1\\2&m-2\end{array}\right|=(m-1)(m-2)-2*(-1)=m^{2}-2m-m+2+2$$=m^{2}-3m+4$
Rezolvam ecuatia:
m²-3m+4=0
Δ=(-3)²-4*1*4=9-16=-7
Cum Δ<0, ecuatia nu are solutii reale.
Daca ecuatia nu are solutii reala, inseamna ca oricare ar fi m din R, det(A(m))≠0, adica matricea A(m) este inversabila, oricare ar fi m din R.

Answer 2

a)
A(0)=(-1 -1)
         (2   -2)   =2-(-2)=2+2=4


c) det (A(m))=m²-3m+2+2=m²-3m+4 care areΔ=9-16=-7<0 deci are peste tot semnul lui1>0 , nu aretradacini, nu se anuleaza
det(A(m)≠0,∀m∈R⇔A este inversabila


b) A(1+m)= (m -1)
                   (2  m-1)

A(1-m)= (-m  -1
              (2    -m-1)


A(1+m)+A(1-m)= (0 -2)  = 2( 0 -1
                             (4 -2)       (2 -1)  ADEVARAT