## MATH1014 Quizzes

Quiz 8: Inverses of matrices
Question 1 Questions
Which of the following is the inverse of the matrix $A=\left[\begin{array}{cc}\hfill 3\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 2\hfill \end{array}\right]$? Exactly one option must be correct)
 a) $\left[\begin{array}{cc}\hfill \frac{1}{3}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2}\hfill \end{array}\right]$ b) $\left[\begin{array}{cc}\hfill \frac{1}{2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{3}\hfill \end{array}\right]$ c) $\left[\begin{array}{cc}\hfill -3\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -2\hfill \end{array}\right]$ d) $\left[\begin{array}{cc}\hfill 2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 3\hfill \end{array}\right]$ e) $\left[\begin{array}{cc}\hfill -2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -3\hfill \end{array}\right]$ f) $\left[\begin{array}{cc}\hfill 0\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 0\hfill \end{array}\right]$

Choice (a) is correct!
Choice (b) is incorrect
Check by multiplying this answer by $A$. The result is not the $2×2$ identity matrix.
Choice (c) is incorrect
Check by multiplying this answer by $A$. The result is not the $2×2$ identity matrix.
Choice (d) is incorrect
Check by multiplying this answer by $A$. The result is not the $2×2$ identity matrix.
Choice (e) is incorrect
Check by multiplying this answer by $A$. The result is not the $2×2$ identity matrix.
Choice (f) is incorrect
Check by multiplying this answer by $A$. The result is not the $2×2$ identity matrix.
Which of the following is the inverse of the matrix $A=\left[\begin{array}{cc}\hfill -5\hfill & \hfill 7\hfill \\ \hfill 3\hfill & \hfill -4\hfill \end{array}\right]$? Exactly one option must be correct)
 a) $\left[\begin{array}{cc}\hfill \frac{-1}{5}\hfill & \hfill \frac{1}{7}\hfill \\ \hfill \frac{1}{3}\hfill & \hfill \frac{-1}{4}\hfill \end{array}\right]$ b) $\left[\begin{array}{cc}\hfill 5\hfill & \hfill -7\hfill \\ \hfill -3\hfill & \hfill 4\hfill \end{array}\right]$ c) $\left[\begin{array}{cc}\hfill -4\hfill & \hfill 7\hfill \\ \hfill 3\hfill & \hfill -5\hfill \end{array}\right]$ d) $\left[\begin{array}{cc}\hfill -4\hfill & \hfill -7\hfill \\ \hfill -3\hfill & \hfill -5\hfill \end{array}\right]$ e) $\left[\begin{array}{cc}\hfill 4\hfill & \hfill 7\hfill \\ \hfill 3\hfill & \hfill 5\hfill \end{array}\right]$ f) The inverse of $A$ does not exist.

Choice (a) is incorrect
Check by multiplying this answer by $A$. The result is not the $2×2$ identity matrix.
Choice (b) is incorrect
Check by multiplying this answer by $A$. The result is not the $2×2$ identity matrix.
Choice (c) is incorrect
Check by multiplying this answer by $A$. The result is not the $2×2$ identity matrix.
Choice (d) is incorrect
Check by multiplying this answer by $A$. The result is not the $2×2$ identity matrix.
Choice (e) is correct!
Choice (f) is incorrect
The inverse does exist! Try again.
Suppose that $A$ and $B$ are $2×2$ matrices and that $AB=BA=\left[\begin{array}{cc}\hfill 2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 2\hfill \end{array}\right].$ Which of the following statements are correct? (More than one option may be correct.) (Zero or more options can be correct)
 a) ${A}^{-1}=B$ b) ${B}^{-1}=A$ c) ${A}^{-1}=\frac{1}{2}B$ d) ${B}^{-1}=\frac{1}{2}A$ e) ${A}^{-1}=2B$ f) ${B}^{-1}=2A$ g) Neither ${A}^{-1}$ nor ${B}^{-1}$ exist.

There is at least one mistake.
For example, choice (a) should be False.
Note that $AB=2{I}_{2}$.
There is at least one mistake.
For example, choice (b) should be False.
Note that $AB=2{I}_{2}$.
There is at least one mistake.
For example, choice (c) should be True.
There is at least one mistake.
For example, choice (d) should be True.
There is at least one mistake.
For example, choice (e) should be False.
Note that $\left(2B\right)A=4{I}_{2}$.
There is at least one mistake.
For example, choice (f) should be False.
Note that $\left(2A\right)B=4{I}_{2}$.
There is at least one mistake.
For example, choice (g) should be False.
Both inverses exist. Try again.
Correct!
1. False Note that $AB=2{I}_{2}$.
2. False Note that $AB=2{I}_{2}$.
3. True
4. True
5. False Note that $\left(2B\right)A=4{I}_{2}$.
6. False Note that $\left(2A\right)B=4{I}_{2}$.
7. False Both inverses exist. Try again.
A sequence of elementary row operations transforms the augmented matrix $\left[A|I\right]$ into
$\left[\begin{array}{cccccc}\hfill 1\hfill & \hfill 3\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 1\hfill \end{array}\right].$
Find ${A}^{-1}$. Exactly one option must be correct)
 a) $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 2\hfill & \hfill 3\hfill & \hfill 1\hfill \end{array}\right]$ b) $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 3\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill \hfill \end{array}\right]$ c) $\left[\begin{array}{ccc}\hfill 10\hfill & \hfill 20\hfill & \hfill 3\hfill \\ \hfill -3\hfill & \hfill -6\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 3\hfill & \hfill 1\hfill \end{array}\right]$ d) ${A}^{-1}$ is undefined. e) $\left[\begin{array}{ccc}\hfill -2\hfill & \hfill 2\hfill & \hfill -3\hfill \\ \hfill -3\hfill & \hfill -6\hfill & \hfill 0\hfill \\ \hfill -3\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$

Choice (a) is incorrect
You still need to perform two operations in order to reduce the left hand matrix to the identity matrix.
Choice (b) is incorrect
This is the left hand matrix. It must be reduced to the identity, and then the right hand matrix is the inverse.
Choice (c) is correct!
Choice (d) is incorrect
The inverse is defined. Try again.
Choice (e) is incorrect
The two operations that need to be performed are ${R}_{2}\to {R}_{2}-2{R}_{3}$ and ${R}_{1}\to {R}_{1}-3{R}_{2}$, in that order.
Let $A=\left[\begin{array}{ccc}\hfill -3\hfill & \hfill -1\hfill & \hfill -2\hfill \\ \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill \\ \hfill 1\hfill & \hfill 4\hfill & \hfill 5\hfill \end{array}\right]$. The inverse of $A$ is: Exactly one option must be correct)
 a) undefined b) $B=\left[\begin{array}{ccc}\hfill -3\hfill & \hfill -1\hfill & \hfill -2\hfill \\ \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill \\ \hfill 1\hfill & \hfill 4\hfill & \hfill 5\hfill \end{array}\right]$ c) $C=\left[\begin{array}{ccc}\hfill -\frac{1}{3}\hfill & \hfill 1\hfill & \hfill -\frac{1}{2}\hfill \\ \hfill \hfill \\ \hfill \frac{1}{2}\hfill & \hfill \frac{1}{3}\hfill & \hfill \frac{1}{4}\hfill \\ \hfill \hfill \\ \hfill 1\hfill & \hfill \frac{1}{4}\hfill & \hfill \frac{1}{5}\hfill \end{array}\right]$ d) $D=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill -6\hfill & \hfill 17\hfill \\ \hfill 4\hfill & \hfill -2\hfill & \hfill 1\hfill \\ \hfill 11\hfill & \hfill -8\hfill & \hfill 0\hfill \end{array}\right]$ e) $E=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 3\hfill & \hfill -2\hfill \\ \hfill 6\hfill & \hfill 13\hfill & \hfill -8\hfill \\ \hfill -5\hfill & \hfill -11\hfill & \hfill 7\hfill \end{array}\right]$

Choice (a) is incorrect
The inverse exists. Try again.
Choice (b) is incorrect
Multiply $A$ by $B$. The result is not the $3×3$ identity matrix.
Choice (c) is incorrect
Multiply $A$ by $C$. The result is not the $3×3$ identity matrix.
Choice (d) is incorrect
Multiply $A$ by $D$. The result is not the $3×3$ identity matrix.
Choice (e) is correct!
Let $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]$ and $B=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$. Which of the following are correct? (Zero or more options can be correct)
 a) $AB=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ b) $BA=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ c) ${A}^{-1}=B$ d) ${B}^{-1}=A$ e) $A$ is not invertible. f) $B$ is not invertible.

There is at least one mistake.
For example, choice (a) should be True.
There is at least one mistake.
For example, choice (b) should be False.
There is at least one mistake.
For example, choice (c) should be False.
There is at least one mistake.
For example, choice (d) should be False.
There is at least one mistake.
For example, choice (e) should be True.
There is at least one mistake.
For example, choice (f) should be True.
Correct!
1. True
2. False
3. False
4. False
5. True
6. True
Let $A=\left[\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right]$ and suppose that ${A}^{-1}=\left[\begin{array}{cc}\hfill 5\hfill & \hfill -2\hfill \\ \hfill -3\hfill & \hfill 1\hfill \end{array}\right]$. Consider the system: $\begin{array}{llll}\hfill ax+by& =2\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill cx+dy& =7\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$ Which of the following is true ? Exactly one option must be correct)
 a) The system has the unique solution $x=-4$, $y=1$. b) The system has no solution. c) The system has infinitely many solutions. d) The system has the unique solution $x=24$, $y=13$. e) It is not possible to say anything about the solution unless we know the values of $a$, $b$, $c$ and $d$.

Choice (a) is correct!
Choice (b) is incorrect
A solution exists. Multiply ${A}^{-1}$ by $\left[\begin{array}{c}\hfill 2\hfill \\ \hfill 7\hfill \end{array}\right]$.
Choice (c) is incorrect
The fact that ${A}^{-1}$ exists tells you that there is a unique solution.
Choice (d) is incorrect
The solution is unique, but these are incorrect values for $x$ and $y$.
Choice (e) is incorrect
The solution is found by multiplying ${A}^{-1}$ by $\left[\begin{array}{c}\hfill -4\hfill \\ \hfill 1\hfill \end{array}\right]$.
Find $x$ if $Ax=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \\ \hfill -1\hfill \end{array}\right]$ and ${A}^{-1}=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill -2\hfill & \hfill 5\hfill \\ \hfill 1\hfill & \hfill -3\hfill & \hfill 6\hfill \\ \hfill -1\hfill & \hfill 3\hfill & \hfill -7\hfill \end{array}\right]$. Exactly one option must be correct)
 a) It is not possible to find $x$ without knowing what $A$ is. b) $x=\left[\begin{array}{c}\hfill -6\hfill \\ \hfill -8\hfill \\ \hfill 9\hfill \end{array}\right]$ c) $x=\left[\begin{array}{c}\hfill 6\hfill \\ \hfill -8\hfill \\ \hfill -5\hfill \end{array}\right]$ d) $x=\left[\begin{array}{c}\hfill -6\hfill \\ \hfill -8\hfill \\ \hfill 11\hfill \end{array}\right]$ e) The system $Ax=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \\ \hfill -1\hfill \end{array}\right]$ is inconsistent. There are no solutions for $x$.

Choice (a) is incorrect
$x$ is found by multiplying ${A}^{-1}$ by $\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \\ \hfill -1\hfill \end{array}\right].$
Choice (b) is correct!
Choice (c) is incorrect
Try again. ($x$ is found by multiplying ${A}^{-1}$ by $\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \\ \hfill -1\hfill \end{array}\right].$)
Choice (d) is incorrect
Try again. ($x$ is found by multiplying ${A}^{-1}$ by $\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \\ \hfill -1\hfill \end{array}\right].$)
Choice (e) is incorrect
There is a solution, given by $x={A}^{-1}\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \\ \hfill -1\hfill \end{array}\right]$
Suppose that $A$ is an invertible matrix, and consider a system of linear equations $Ax=b$. Which of the following statements are true? (Zero or more options can be correct)
 a) The system has a unique solution. b) The system has infinitely many solutions. c) The system has no solutions. d) It depends on $A$ and $b$. Any of (a), (b) or (c) could be true. e) If $b=0$ the only solution is the trivial solution. f) If $b=0$ the system has infinitely many solutions.

There is at least one mistake.
For example, choice (a) should be True.
There is at least one mistake.
For example, choice (b) should be False.
There is at least one mistake.
For example, choice (c) should be False.
There is at least one mistake.
For example, choice (d) should be False.
There is at least one mistake.
For example, choice (e) should be True.
There is at least one mistake.
For example, choice (f) should be False.
Correct!
1. True
2. False
3. False
4. False
5. True
6. False
Suppose that $A$ is a square matrix, and that ${A}^{-1}$ does not exist. Which of the following statements are true? (Zero or more options can be correct)
 a) Any system of linear equations $Ax=b$ will have a unique solution. b) Any system of linear equations $Ax=b$ will have infinitely many solutions. c) Any system of linear equations $Ax=b$ will be inconsistent. d) A system of linear equations $Ax=b$ could have infinitely many solutions, or no solution, depending on $b$. e) A system of linear equations $Ax=0$ will have infinitely many solutions. f) A system of linear equations $Ax=0$ will have no solution.

There is at least one mistake.
For example, choice (a) should be False.
There is at least one mistake.
For example, choice (b) should be False.
There is at least one mistake.
For example, choice (c) should be False.
There is at least one mistake.
For example, choice (d) should be True.
There is at least one mistake.
For example, choice (e) should be True.
There is at least one mistake.
For example, choice (f) should be False.
Homogeneous equations always have at least one solution.
Correct!
1. False
2. False
3. False
4. True
5. True
6. False Homogeneous equations always have at least one solution.