Eigenvalues & Eigenvector

If A is a square matrix Ax=λx for nonzero scalar λ, then x is an eigenvector of A

• True
• False

If λ is an eigenvalue of matrix A, then linear system (A-λI)v=0 has only trivial solution.

• True
• False

If λ is an eigenvalue of matrix A, then the eigenspace of A corresponding to λ is the set of eigenvectors of A corresponding to λ

• True
• False

If 0 is an eigenvalue of a matrix A, then A²  is singular

• True
• False

The eigenvalues of matrix A are the same as the eigenvalues of RREF of A

• True
• False

If 0 is an eigenvalue of matrix A, then the set of columns of A is linearly independent

• True
• False

Find eigenvalues for matrix A

• λ=1,3
• λ=-1,2
• λ=1,-2
• λ=1,4

Find the eigenvectors for matrix A corresponding to the smallest eigenvalue.

• v={r(0,2),r∊R, r≠0}
• v={r(1,0),r∊R, r≠0}
• v={r(0,1),r∊R, r≠0}
• v={r(1,2),r∊R, r≠0}

Find the dimension of eigenspace/geometric multiplicity for matrix A.

• 1
• 2
• 3
• 4

Find the eigenvalues for A⁹ of matrix A

• λ=1,1/8,512
• λ=1,1/2,2
• λ=1,1/512,512
• λ=1,0,512