Respuesta :
[tex]A^{2} u is equal to 80 v_{1}+9 v_{2}[/tex]
Answer: Option C
Step-by-step explanation:
Given A is [tex]2 \times 2[/tex] matrix has Eigen values [tex]\lambda_{1}=4[/tex] and \lambda_{2}=3 with Eigen vectors [tex]v_{1} \text { and } v_{2}[/tex] respectively.
[tex]\lambda_{1}=4[/tex] and [tex]v_{1}[/tex] is the eigen vector, substitute this to A so then
[tex]A v_{1}=\lambda_{1} v_{1}=4 v_{1}[/tex]
Squaring ‘A’ value ‘4’, we get
[tex]A^{2} v_{1}=16 v_{1}[/tex]
Given [tex]u=5 v_{1}+v_{2}[/tex], from this, the above can be written as
[tex]A^{2}\left(5 v_{1}\right)=5 A^{2} v_{1}=5 \times 16 v_{1}=80 v_{1}[/tex]
Similarly, [tex]\lambda_{2}=3[/tex] and [tex]v_{2}[/tex] is the eigen vector. Then,
[tex]A v_{2}=\lambda_{2} v_{2}=3 \times v_{2}[/tex]
Squaring ‘A’ value ‘3’, we get
[tex]A^{2} v_{2}=9 \times v_{2}[/tex]
To find [tex]A^{2} u[/tex], multiply [tex]A^{2}[/tex] in both sides of the equation [tex]u=5 v_{1}+v_{2}[/tex], we get
[tex]A^{2} u=A^{2} \times\left(5 v_{1}+v_{2}\right)[/tex]
[tex]A^{2} u=5 A^{2} v_{1}+A^{2} v_{2}[/tex]
Substitute the value that found above,
[tex]A^{2} u=80 v_{1}+9 v_{2}[/tex]
