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 $7^{7}.$

Solution:

Final Answer:

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

Solution:

Final Answer:

3. True or false: If $A$ is a $4\times 4$ matrix with characteristic equation Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lambda (\lambda -1)(\lambda +1)(\lambda +e)=0,} then $A$ is diagonalizable.

Solution:

Final Answer:

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

Solution:

Final Answer:

5. True or false: If $A$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle B} are invertible $n\times n$ matrices, then so is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A+B.}

Solution:

Final Answer:

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 $\mathbb {R} ^{3}.$

Solution:

Final Answer:

7. True or false: Let $C=AB$ for $4\times 4$ matrices $A$ and $B.$ If Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle C} is invertible, then $A$ is invertible.

Solution:

Final Answer:

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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {v}}={\vec {0}}.}

Solution:

Final Answer:

9. True or false: If $A$ is an invertible $3\times 3$ matrix, and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle B} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle C} are $3\times 3$ matrices such that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle AB=AC,} then $B=C.$

Solution:

Final Answer:

10.

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

Solution:

Final Answer:

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

Solution:

Final Answer:

12. Consider the matrix Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 $A.$

Solution:

Final Answer:

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

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

Solution:

Final Answer:

14. Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T({\vec {x}})=B{\vec {x}}.} Is $T$ onto? Explain.

Solution:

Final Answer:

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {u}}=7{\vec {e_{1}}}-4{\vec {e_{2}}}.} Find Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T({\vec {u}}).}

(c) Is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{bmatrix}-1\\3\end{bmatrix}}} in the range of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T?} Explain.

Solution:

Final Answer:

16. Let $A$ and $B$ be $6\times 6$ matrices with det Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A=-10} and det $B=5.$ Use properties of

determinants to compute:

(a) det Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 3A}

(b) det Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (A^{T}B^{-1})}

Solution:

Final Answer:

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.

Solution:

Final Answer:

18. Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {v}}={\begin{bmatrix}-1\\3\\0\end{bmatrix}}} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {y}}={\begin{bmatrix}2\\0\\5\end{bmatrix}}.}

(a) Find a unit vector in the direction of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {v}}.}

(b) Find the distance between ${\vec {v}}$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {y}}.}

(c) Let $L=$SpanFailed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \{{\vec {v}}\}.} Compute the orthogonal projection of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {y}}} onto $L.$

Solution:

Final Answer:

19. Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle W=} SpanFailed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\Bigg \{}{\begin{bmatrix}2\\0\\-1\\0\end{bmatrix}},{\begin{bmatrix}-3\\1\\0\\0\end{bmatrix}}{\Bigg \}}.} Is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\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.

Solution:

Final Answer:

20.

(a) Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 $A-B^{T}.$

Solution:

Final Answer:

21.

Solution:

Final Answer:

22.

Solution:

Final Answer:

23.

Solution:

Final Answer:

24.

Solution:

Final Answer:

25.

Solution:

Final Answer:

26.

Solution:

Final Answer:

27.

Solution:

Final Answer:

28.

Solution:

Final Answer:

29.

Solution:

Final Answer:

30.

Solution:

Final Answer:

31.

Solution:

Final Answer: