Question

Am nevoie doar de punctul c)

Answer 1

$f_3 = \dfrac{1}{n!}\cdot (X+1)(X+2)(X+3)\\ \\ \det\Big(f_3(A)\Big) = \dfrac{1}{6}\det \Big((A+I_3)(A+2I_3)(A+3I_3)\Big)= \\ \\ = \dfrac{1}{6}\det (A+I_3)\cdot \dfrac{1}{6}\det (A+2I_3)\cdot \dfrac{1}{6}\det (A+3I_3) = \\ \\ = \dfrac{1}{6}\det \Big[(I_3-(-1)\cdot A\Big]\cdot \dfrac{1}{6}\det\Big[2(I_3-(\frac{1}{2})\cdot A)\Big] \cdot \\ \cdot \dfrac{1}{6}\det\Big[3(I_3-(\frac{1}{3})\cdot A)\Big] =$

$= \dfrac{1}{6}\cdot 1\cdot \dfrac{1}{6}\cdot 2^3\cdot 1\cdot \dfrac{1}{6}\cdot 3^3\cdot1 = \\ \\ = \dfrac{6^3}{6^3} = 1\\ \\ \\\det(X\cdot Y) = \det(X)\cdot \det(Y) \\ \det(a\cdot A) = a^n\cdot \det(A)\quad (daca~matricea~A~e~de~ordin~n)$