Respuesta :
This statement is false.
We know if Mv is a given matrix with v as the eigenvector corresponding to the eigenvalue k then the following relation holds:
Mv = kv
Also, we know that sum of two vectors [tex]v_{1}[/tex] and [tex]v_{2}[/tex] is given by ([tex]v_{1}[/tex] + [tex]v_{2}[/tex]).
Using it, here we will explain whether the given statement is true or not.
Let us consider the given matrix to be A with the eigenvalues [tex]λ_{1}[/tex] and [tex]λ_{2}[/tex] . Let the corresponding eigenvectors be [tex]v_{1}[/tex] and [tex]v_{2}[/tex] respectively. We get,
A[tex]v_{1}[/tex] =[tex]λ_{1}[/tex][tex]v_{1}[/tex] , A[tex]v_{2}[/tex] = [tex]λ_{2}[/tex][tex]v_{2}[/tex]
Now, we calculate,
A ([tex]v_{1} + v_{2}[/tex] ) = A[tex]v_{1} + Av_{2}[/tex]
= [tex]λ_{1} v_{1} + λ_{2}v_{2}[/tex]
≠[tex]k ( v_{1} + v_{2} )[/tex]
This shows that [tex](v_{1} + v_{2} )[/tex] is not the eigenvector of matrix A.
Hence, the given statement is - false.
Read more about the eigenvector :
https://brainly.com/question/14406741
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