031 Review Part 3

These questions are from sample exams and actual exams at other universities. The questions are meant to represent the material usually covered in Math 31 for the final. An actual test may or may not be similar.

Click on the  boxed problem numbers  to go to a solution.

 Problem 1 

(a) Is the matrix  $A={\begin{bmatrix}3&1\\0&3\end{bmatrix}}$  diagonalizable? If so, explain why and diagonalize it. If not, explain why not.

(b) Is the matrix  $A={\begin{bmatrix}2&0&-2\\1&3&2\\0&0&3\end{bmatrix}}$  diagonalizable? If so, explain why and diagonalize it. If not, explain why not.

 Problem 2 

Find the eigenvalues and eigenvectors of the matrix  $A={\begin{bmatrix}1&1&1\\0&-1&1\\0&0&2\end{bmatrix}}.$

 Problem 3 

Let  $A={\begin{bmatrix}5&1\\0&5\end{bmatrix}}.$

(a) Find a basis for the eigenspace(s) of  $A.$

(b) Is the matrix  $A$  diagonalizable? Explain.

 Problem 4 

Let  $W={\text{Span }}{\Bigg \{}{\begin{bmatrix}2\\0\\-1\\0\end{bmatrix}},{\begin{bmatrix}-3\\1\\0\\0\end{bmatrix}}{\Bigg \}}.$  Is  ${\begin{bmatrix}2\\6\\4\\0\end{bmatrix}}$  in  $W^{\perp }?$  Explain.

 Problem 5 

Find a formula for  ${\begin{bmatrix}1&-6\\2&-6\end{bmatrix}}^{k}$  by diagonalizing the matrix.

 Problem 6 

(a) Show that if  ${\vec {x}}$  is an eigenvector of the matrix  $A$  corresponding to the eigenvalue 2, then  ${\vec {x}}$  is an eigenvector of  $A^{3}-A^{2}+I.$  What is the corresponding eigenvalue?

(b) Show that if  ${\vec {y}}$  is an eigenvector of the matrix  $A$  corresponding to the eigenvalue 3 and  $A$  is invertible, then  ${\vec {y}}$  is an eigenvector of  $A^{-1}.$  What is the corresponding eigenvalue?

 Problem 7 

Let  $A={\begin{bmatrix}3&0&-1\\0&1&-3\\1&0&0\end{bmatrix}}{\begin{bmatrix}3&0&0\\0&4&0\\0&0&3\end{bmatrix}}{\begin{bmatrix}0&0&1\\-3&1&9\\-1&0&3\end{bmatrix}}.$

Use the Diagonalization Theorem to find the eigenvalues of  $A$  and a basis for each eigenspace.

 Problem 8 

Give an example of a  $3\times 3$  matrix  $A$  with eigenvalues 5,-1 and 3.

 Problem 9 

Assume  $A^{2}=I.$  Find  ${\text{Nul }}A.$

 Problem 10 

Show that if  ${\vec {x}}$  is an eigenvector of the matrix product  $AB$  and  $B{\vec {x}}\neq {\vec {0}},$  then  $B{\vec {x}}$  is an eigenvector of  $BA.$

 Problem 11 

Suppose  $\{{\vec {u}},{\vec {v}}\}$  is a basis of the eigenspace corresponding to the eigenvalue 0 of a  $5\times 5$  matrix  $A.$

(a) Is  ${\vec {w}}={\vec {u}}-2{\vec {v}}$  an eigenvector of  $A?$  If so, find the corresponding eigenvalue.

If not, explain why.

(b) Find the dimension of  ${\text{Col }}A.$