031 Review Problems

This is a list of sample problems and is meant to represent the material usually covered in Math 31. An actual test may or may not be similar.

1. True or false: If all the entries of a  $7\times 7$  matrix  $A$  are  $7,$  then  ${\text{det }}A$  must be  $7^{7}.$

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2. True or false: If a matrix  $A^{2}$  is diagonalizable, then the matrix  $A$  must be diagonalizable as well.

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3. True or false: If  $A$  is a  $4\times 4$  matrix with characteristic equation  $\lambda (\lambda -1)(\lambda +1)(\lambda +e)=0,$  then  $A$  is diagonalizable.

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4. True or false: If  $A$  is invertible, then  $A$  is diagonalizable.

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5. True or false: If  $A$  and  $B$  are invertible  $n\times n$  matrices, then so is  $A+B.$

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6. True or false: If  $A$  is a  $3\times 5$  matrix and  ${\text{dim Nul }}A=2,$  then  $A{\vec {x}}={\vec {b}}$  is consistent for all  ${\vec {b}}$  in  $\mathbb {R} ^{3}.$

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7. True or false: Let  $C=AB$  for  $4\times 4$  matrices  $A$  and  $B.$  If  $C$  is invertible, then  $A$  is invertible.

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8. True or false: Let  $W$  be a subspace of  $\mathbb {R} ^{4}$  and  ${\vec {v}}$  be a vector in  $\mathbb {R} ^{4}.$  If  ${\vec {v}}\in W$  and  ${\vec {v}}\in W^{\perp },$  then  ${\vec {v}}={\vec {0}}.$

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9. True or false: If  $A$  is an invertible  $3\times 3$  matrix, and  $B$  and  $C$  are  $3\times 3$  matrices such that  $AB=AC,$  then  $B=C.$

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10.

(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.

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

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12. Consider the matrix  $A={\begin{bmatrix}1&-4&9&-7\\-1&2&-4&1\\5&-6&10&7\end{bmatrix}}$  and assume that it is row equivalent to the matrix

$B={\begin{bmatrix}1&0&-1&5\\0&-2&5&-6\\0&0&0&0\end{bmatrix}}.$

(a) List rank  $A$  and  ${\text{dim Nul }}A.$

(b) Find bases for  ${\text{Col }}A$  and  ${\text{Nul }}A.$  Find an example of a nonzero vector that belongs to  ${\text{Col }}A,$  as well as an example of a nonzero vector that belongs to  ${\text{Nul }}A.$

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13. Find the dimension of the subspace spanned by the given vectors. Are these vectors linearly independent?

${\begin{bmatrix}1\\0\\2\end{bmatrix}},{\begin{bmatrix}3\\1\\1\end{bmatrix}},{\begin{bmatrix}-2\\-1\\1\end{bmatrix}},{\begin{bmatrix}5\\2\\2\end{bmatrix}}$

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14. Let  $B={\begin{bmatrix}1&-2&3&4\\0&3&0&0\\0&5&1&2\\0&-1&3&6\end{bmatrix}}.$

(a) Is  $B$  invertible? Explain.

(b) Define a linear transformation  $T$  by the formula  $T({\vec {x}})=B{\vec {x}}.$  Is  $T$  onto? Explain.

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15. Suppose  $T$  is a linear transformation given by the formula

$T{\Bigg (}{\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\\\end{bmatrix}}{\Bigg )}={\begin{bmatrix}5x_{1}-2.5x_{2}+10x_{3}\\-x_{1}+0.5x_{2}-2x_{3}\end{bmatrix}}$

(a) Find the standard matrix for  $T.$

(b) Let  ${\vec {u}}=7{\vec {e_{1}}}-4{\vec {e_{2}}}.$  Find  $T({\vec {u}}).$

(c) Is  ${\begin{bmatrix}-1\\3\end{bmatrix}}$  in the range of  $T?$  Explain.

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16. Let  $A$  and  $B$  be  $6\times 6$  matrices with  ${\text{det }}A=-10$  and  ${\text{det }}B=5.$  Use properties of determinants to compute:

(a)  ${\text{det }}3A$

(b)  ${\text{det }}(A^{T}B^{-1})$

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17. 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.

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18. Let  ${\vec {v}}={\begin{bmatrix}-1\\3\\0\end{bmatrix}}$  and  ${\vec {y}}={\begin{bmatrix}2\\0\\5\end{bmatrix}}.$

(a) Find a unit vector in the direction of  ${\vec {v}}.$

(b) Find the distance between  ${\vec {v}}$  and  ${\vec {y}}.$

(c) Let  $L={\text{Span }}\{{\vec {v}}\}.$  Compute the orthogonal projection of  ${\vec {y}}$  onto  $L.$

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19. 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.

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20.

(a) Let  $T:\mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$  be a transformation given by

$T{\bigg (}{\begin{bmatrix}x\\y\end{bmatrix}}{\bigg )}={\begin{bmatrix}1-xy\\x+y\end{bmatrix}}.$

Determine whether  $T$  is a linear transformation. Explain.

(b) Let  $A={\begin{bmatrix}1&-3&0\\-4&1&1\end{bmatrix}}$  and  $B={\begin{bmatrix}2&1\\1&0\\-1&1\end{bmatrix}}.$  Find  $AB,~BA^{T}$  and  $A-B^{T}.$

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21. Let  $A={\begin{bmatrix}1&3&8\\2&4&11\\1&2&5\end{bmatrix}}.$  Find  $A^{-1}$  if possible.

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

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23.

(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?

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24. 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.

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

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

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27. If  $A$  is an  $n\times n$  matrix such that  $AA^{T}=I,$  what are the possible values of  ${\text{det }}A?$

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28. 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  $BA.$

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29.

(a) Suppose a  $6\times 8$  matrix  $A$  has 4 pivot columns. What is  ${\text{dim Nul }}A?$  Is  ${\text{Col }}A=\mathbb {R} ^{4}?$  Why or why not?

(b) If  $A$  is a  $7\times 5$  matrix, what is the smallest possible dimension of  ${\text{Nul }}A?$

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30. Consider the following system of equations.

$x_{1}+kx_{2}=1$

$3x_{1}+5x_{2}=2k$

Find all real values of  $k$  such that the system has only one solution.

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31. 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  ${\text{Col }}A.$

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