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 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 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 dim Nul $A.$

(b) Find bases for Col $A$ and Nul $A.$ Find an example of a nonzero vector that belongs to Col $A,$ as well as an example of a nonzero vector that belongs to 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 det $A=-10$ and det $B=5.$ Use properties of

determinants to compute:

(a) det $3A$

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

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19. Let $W=$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 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 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 dim Nul $A?$ Is 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 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 Col $A.$

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