I'm preparing final exam on linear algebra and come across with these problem:

#1

Let A be an n*n matrix.Assume that for some , and .

a)Find all the eigenvalues of A.

b)Prove that A is not diagonalizable.

c)Prove that k<n

d)If k=n-1,find the dimension of null A

#2

Let A be an n*n matrix and be two real numbers.Assume that

Prove that A is diagonalizable.

#3

Let and be eigenvectors correspond to distinct eigenvalues of an n*n matrix A.

Prove that is not an eigenvector of A.

For #3,

Suppose on the contrary, is an eigenvector of A

Then there exists a number n such that

Since eigenvectors are lin. ind.

therefore,

Contradict to are distinct.

Is it right ?