031 Review Part 3

This is a sample, and is meant to represent the material usually covered in Math 9C 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  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A={\begin{bmatrix}5&1\\0&5\end{bmatrix}}.}

(a) Find a basis for the eigenspace(s) of  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A.}

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

 Problem 4 

Let  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle W^{\perp }?}   Explain.

 Problem 5 

Find a formula for  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\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  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {y}}}   is an eigenvector of the matrix  $A$  corresponding to the eigenvalue 3 and  $A$  is invertible, then  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {y}}}   is an eigenvector of  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A^{-1}.}   What is the corresponding eigenvalue?

 Problem 7 

Let  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 3\times 3}   matrix  $A$  with eigenvalues 5,-1 and 3.

 Problem 9 

Assume  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A^{2}=I.}   Find  ${\text{Nul }}A.$

 Problem 10 

Show that if  ${\vec {x}}$  is an eigenvector of the matrix product  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle AB}   and  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle B{\vec {x}}\neq {\vec {0}},}   then  $B{\vec {x}}$  is an eigenvector of  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle BA.}

 Problem 11 

Suppose  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \{{\vec {u}},{\vec {v}}\}}   is a basis of the eigenspace corresponding to the eigenvalue 0 of a  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 5\times 5}   matrix  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A.}

(a) Is  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {w}}={\vec {u}}-2{\vec {v}}}   an eigenvector of  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A?}   If so, find the corresponding eigenvalue.

If not, explain why.

(b) Find the dimension of  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\text{Col }}A.}