I'm preparing final exam on linear algebra and come across with these problem:
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
Let A be an n*n matrix and be two real numbers.Assume that
Prove that A is diagonalizable.
Let and be eigenvectors correspond to distinct eigenvalues of an n*n matrix A.
Prove that is not an eigenvector of A.
Suppose on the contrary, is an eigenvector of A
Then there exists a number n such that
Since eigenvectors are lin. ind.
Contradict to are distinct.
Is it right ?