Question

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

Factors and multiples
ANSWERED
asked 2021-08-17
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.

Expert Answers (1)

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.
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