Question # Suppose \lambda is an eigenvalue of the n \times n

Factors and multiples
ANSWERED Suppose $$\displaystyle\lambda$$ is an eigenvalue of the $$\displaystyle{n}\times{n}$$ matrix A. If u and v are eigenvectors of A corresponding to $$\displaystyle\lambda$$, which of the following statement MUST be false?
1)$$\displaystyle\lambda={0}$$
2) u and v are scalars multiple of each other.
3) nullity $$\displaystyle{\left({A}-\lambda{I}\right)}={0}$$
4) $$\displaystyle{7}{v}-{3}{u}$$ is also an eigenvector of A. 2021-08-18
Step 1
In the question we have to find falsa statement.
Step 2
Result - option d
Given that u and v are eigen vectors of $$\displaystyle{A}{\left({n}\times{n}\right)}$$ matrix.
a) $$\displaystyle\lambda={0}$$ - true eigen values can be zero or non zero or positive or negative too.
b) u and v are scalar multiple of each other - true because eigen vactor are written as [1,3]t or any quantity which implies eigen vectors are scalar multiples.
c) nullity $$\displaystyle{A}-\lambda{I}={0}$$ - true as eigen values are found by putting eigen values in this form.
d) $$\displaystyle{7}{v}-{3}{u}$$ is also a eigen vector of matrix A - false because eigen vectors can not be calculated by taking sum, difference or any calculus.