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)

*Choice (a) is correct!*

*Choice (b) is incorrect*

Check by multiplying
this answer by $A$. The
result is not the $2\times 2$
identity matrix.

*Choice (c) is incorrect*

Check by multiplying
this answer by $A$. The
result is not the $2\times 2$
identity matrix.

*Choice (d) is incorrect*

Check by multiplying
this answer by $A$. The
result is not the $2\times 2$
identity matrix.

*Choice (e) is incorrect*

Check by multiplying
this answer by $A$. The
result is not the $2\times 2$
identity matrix.

*Choice (f) is incorrect*

Check by multiplying
this answer by $A$. The
result is not the $2\times 2$
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)

*Choice (a) is incorrect*

Check by multiplying
this answer by $A$. The
result is not the $2\times 2$
identity matrix.

*Choice (b) is incorrect*

Check by multiplying this
answer by $A$. The result is
not the $2\times 2$ identity matrix.

*Choice (c) is incorrect*

Check by multiplying
this answer by $A$. The
result is not the $2\times 2$
identity matrix.

*Choice (d) is incorrect*

Check by multiplying
this answer by $A$. The
result is not the $2\times 2$
identity matrix.

*Choice (e) is correct!*

*Choice (f) is incorrect*

The inverse does exist! Try again.

Suppose that $A$
and $B$ are
$2\times 2$ 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)

For example, choice (a) should be False.

For example, choice (b) should be False.

For example, choice (c) should be True.

For example, choice (d) should be True.

For example, choice (e) should be False.

For example, choice (f) should be False.

For example, choice (g) should be False.

*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!*

*False*Note that $AB=2{I}_{2}$.*False*Note that $AB=2{I}_{2}$.*True**True**False*Note that $\left(2B\right)A=4{I}_{2}$.*False*Note that $\left(2A\right)B=4{I}_{2}$.*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)
*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)

*Choice (a) is incorrect*

The inverse exists. Try again.

*Choice (b) is incorrect*

Multiply
$A$ by
$B$. The result
is not the $3\times 3$
identity matrix.

*Choice (c) is incorrect*

Multiply
$A$ by
$C$. The result
is not the $3\times 3$
identity matrix.

*Choice (d) is incorrect*

Multiply
$A$ by
$D$. The result
is not the $3\times 3$
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)

For example, choice (a) should be True.

For example, choice (b) should be False.

For example, choice (c) should be False.

For example, choice (d) should be False.

For example, choice (e) should be True.

For example, choice (f) should be True.

*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!*

*True**False**False**False**True**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)

*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)

*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)

For example, choice (a) should be True.

For example, choice (b) should be False.

For example, choice (c) should be False.

For example, choice (d) should be False.

For example, choice (e) should be True.

For example, choice (f) should be False.

*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!*

*True**False**False**False**True**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)

For example, choice (a) should be False.

For example, choice (b) should be False.

For example, choice (c) should be False.

For example, choice (d) should be True.

For example, choice (e) should be True.

For example, choice (f) should be False.

*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!*

*False**False**False**True**True**False*Homogeneous equations always have at least one solution.