Question

punctul c stiti sa il faceti?

Answer 1

Din a) avem ca:

A^2=2A-I2

A^3=A^2*A=(2A-I2)*A=2A^2-A=2(2A-I2)-A=4A-2I2-A=3A-2I2
A^4=A^3*A=(3A-2I2)*A=3A^2-2A=3(2A-I2)-2A=6A-3I2-2A=4A-3I2

P(n): A^n=nA-(n-1)I2

Demonstram prin iductie ca P(n) este adevarata.
1) Verificam P(2): A^2=2A-I2 -adevarat 
verificat anterior
2) Presupunem P(k): A^k=kA-(k-1)I2 -adevarat
Demonstram P(k+1): A^(k+1)=(k+1)A-kI2 -adevarat
A^(k+1)=A^k*A=[kA-(k-1)I2]*A=kA^2-(k-1)A=k(2A-I2)-(k-1)A=2kA-kI2-kA+A=A(2k-k+1)-kI2=(k+1)A-kI2
Deci: P(k+1): A^(k+1)=(k+1)A-kI2 -adevarat
Din 1) si 2), conform principiului inductiei matematice, avem ca:
P(n): A^n=nA-(n-1)I2 -adevarat.

Daca n=30 => A^30=30A-29I2

Answer 2

$A^{30}= \left[\begin{array}{cc}4&3&-3&-2\end{array}\right] ^{30} = \left[\begin{array}{cc}91&90&-90&-89\end{array}\right]$