What is the size of the matrix $A=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill \\ \hfill 12\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 1\hfill \\ \hfill 13\hfill & \hfill 14\hfill & \hfill 1\hfill & \hfill 2\hfill \end{array}\right]$ ?
Exactly one option must be correct)

*Choice (a) is incorrect*

Try again. The size of a matrix is the number of rows
$\times $ the
number of columns.

*Choice (b) is incorrect*

Try again. The size of a matrix is the number of rows
$\times $ the
number of columns.

*Choice (c) is correct!*

$A$ has 3 rows and 4
columns and hence it is a $3\times 4$
matrix.

*Choice (d) is incorrect*

Try again. The size of a matrix is the number of rows
$\times $ the
number of columns.

What is the $\left(2,3\right)$ entry
in the matrix $A=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill \\ \hfill 12\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 1\hfill \\ \hfill 13\hfill & \hfill 14\hfill & \hfill 1\hfill & \hfill 2\hfill \end{array}\right]$ ?
Exactly one option must be correct)

*Choice (a) is incorrect*

Try again. The $\left(2,3\right)$
entry is in row 2 and column 3.

*Choice (b) is correct!*

We are looking for the entry in the second row and the third column of
$A$,
which is 4.

*Choice (c) is incorrect*

Try again. The $\left(2,3\right)$
entry is in row 2 and column 3.

*Choice (d) is incorrect*

Try again. The $\left(2,3\right)$
entry is in row 2 and column 3.

Which sums can be made from the following matrices ?

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 False.

For example, choice (f) should be True.

For example, choice (g) should be True.

For example, choice (h) should be False.

For example, choice (i) should be True.

For example, choice (j) should be False.

For example, choice (k) should be False.

For example, choice (l) should be False.

For example, choice (m) should be True.

For example, choice (n) should be True.

$$A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 4\hfill & \hfill 5\hfill & \hfill 6\hfill \end{array}\right],B=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 5\hfill & \hfill 6\hfill \end{array}\right],C=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 4\hfill & \hfill 5\hfill & \hfill 6\hfill \\ \hfill 7\hfill & \hfill 8\hfill & \hfill 9\hfill \end{array}\right],D=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 4\hfill \end{array}\right].$$

(Zero or more options can be correct)
*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.

Only
matrices of the same size can be added.

*There is at least one mistake.*

For example, choice (c) should be False.

Only
matrices of the same size can be added.

*There is at least one mistake.*

For example, choice (d) should be False.

Only
matrices of the same size can be added.

*There is at least one mistake.*

For example, choice (e) should be False.

Only
matrices of the same size can be added.

*There is at least one mistake.*

For example, choice (f) should be True.

*There is at least one mistake.*

For example, choice (g) should be True.

*There is at least one mistake.*

For example, choice (h) should be False.

Only
matrices of the same size can be added.

*There is at least one mistake.*

For example, choice (i) should be True.

*There is at least one mistake.*

For example, choice (j) should be False.

Only
matrices of the same size can be added.

*There is at least one mistake.*

For example, choice (k) should be False.

*There is at least one mistake.*

For example, choice (l) should be False.

Only
matrices of the same size can be added.

*There is at least one mistake.*

For example, choice (m) should be True.

*There is at least one mistake.*

For example, choice (n) should be True.

*Correct!*

*True**False*Only matrices of the same size can be added.*False*Only matrices of the same size can be added.*False*Only matrices of the same size can be added.*False*Only matrices of the same size can be added.*True**True**False*Only matrices of the same size can be added.*True**False*Only matrices of the same size can be added.*False**False*Only matrices of the same size can be added.*True**True*

Let $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill -3\hfill & \hfill 4\hfill \\ \hfill 2\hfill & \hfill 10\hfill & \hfill -7\hfill \end{array}\right]$
and $B=\left[\begin{array}{ccc}\hfill 5\hfill & \hfill -2\hfill & \hfill -3\hfill \\ \hfill 0\hfill & \hfill 11\hfill & \hfill 12\hfill \end{array}\right]$.
What is $A+B$?
Exactly one option must be correct)

*Choice (a) is incorrect*

Matrix addition is defined by adding the corresponding entries of the two
matrices.

*Choice (b) is incorrect*

Matrix addition is defined by adding the corresponding entries of the two
matrices.

*Choice (c) is correct!*

*Choice (d) is incorrect*

Matrix addition is defined by adding the corresponding entries of the two matrices.

*Choice (e) is incorrect*

*Choice (f) is incorrect*

Matrix addition is defined by adding the corresponding entries of the two
matrices.

If $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 3\hfill \\ \hfill 2\hfill & \hfill -1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 1\hfill \end{array}\right]$ find the
matrix $7A$.
Exactly one option must be correct)

*Choice (a) is incorrect*

When multiplying a matrix by a scalar, each entry in the matrix is multiplied by the
scalar.

*Choice (b) is incorrect*

When
multiplying a matrix by a scalar, each entry in the matrix is multiplied by the
scalar.

*Choice (c) is incorrect*

When
multiplying a matrix by a scalar, each entry in the matrix is multiplied by the
scalar.

*Choice (d) is correct!*

Consider the four matrices

For example, choice (a) should be True.

For example, choice (b) should be True.

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.

For example, choice (g) should be False.

For example, choice (h) should be False.

For example, choice (i) should be True.

For example, choice (j) should be False.

For example, choice (k) should be True.

For example, choice (l) should be True.

For example, choice (m) should be True.

For example, choice (n) should be False.

For example, choice (o) should be True.

For example, choice (p) should be True.

$$A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 4\hfill & \hfill 5\hfill & \hfill 6\hfill \end{array}\right],\phantom{\rule{1em}{0ex}}B=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 4\hfill \\ \hfill 5\hfill & \hfill 6\hfill \end{array}\right],\phantom{\rule{1em}{0ex}}C=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 4\hfill & \hfill 5\hfill & \hfill 6\hfill \\ \hfill 7\hfill & \hfill 8\hfill & \hfill 9\hfill \end{array}\right]\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}D=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 4\hfill \end{array}\right].$$

Which of the following products can be made from these matrices ?
(Zero or more options can be correct)
*There is at least one mistake.*

For example, choice (a) should be True.

$A$ is a
$2\times 3$ matrix hence we can only
post-multiply $A$ by a matrix
with 3 rows and pre-multiply $A$
by a matrix with 2 columns.

*There is at least one mistake.*

For example, choice (b) should be True.

*There is at least one mistake.*

For example, choice (c) should be False.

$A$ is a
$2\times 3$ matrix hence we can only
post-multiply $A$ by a matrix
with 3 rows and pre-multiply $A$
by a matrix with 2 columns.

*There is at least one mistake.*

For example, choice (d) should be False.

$A$ is a
$2\times 3$ matrix hence we can only
post-multiply $A$ by a matrix
with 3 rows and pre-multiply $A$
by a matrix with 2 columns.

*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.

*There is at least one mistake.*

For example, choice (g) should be False.

$B$ is a
$3\times 2$ matrix hence we can only
post-multiply $B$ by a matrix
with 2 rows and pre-multiply $B$
by a matrix with 3 columns.

*There is at least one mistake.*

For example, choice (h) should be False.

$C$ is a
$3\times 3$ matrix hence we can only
post-multiply $C$ by a matrix
with 3 rows and pre-multiply $C$
by a matrix with 3 columns

*There is at least one mistake.*

For example, choice (i) should be True.

*There is at least one mistake.*

For example, choice (j) should be False.

$C$ is a
$3\times 3$ matrix hence we can only
post-multiply $C$ by a matrix
with 3 rows and pre-multiply $C$
by a matrix with 3 columns

*There is at least one mistake.*

For example, choice (k) should be True.

*There is at least one mistake.*

For example, choice (l) should be True.

*There is at least one mistake.*

For example, choice (m) should be True.

*There is at least one mistake.*

For example, choice (n) should be False.

$C$ is a
$3\times 3$ matrix hence we can only
post-multiply $C$ by a matrix
with 3 rows and pre-multiply $C$
by a matrix with 3 columns

*There is at least one mistake.*

For example, choice (o) should be True.

*There is at least one mistake.*

For example, choice (p) should be True.

*Correct!*

*True*$A$ is a $2\times 3$ matrix hence we can only post-multiply $A$ by a matrix with 3 rows and pre-multiply $A$ by a matrix with 2 columns.*True**False*$A$ is a $2\times 3$ matrix hence we can only post-multiply $A$ by a matrix with 3 rows and pre-multiply $A$ by a matrix with 2 columns.*False*$A$ is a $2\times 3$ matrix hence we can only post-multiply $A$ by a matrix with 3 rows and pre-multiply $A$ by a matrix with 2 columns.*True**True**False*$B$ is a $3\times 2$ matrix hence we can only post-multiply $B$ by a matrix with 2 rows and pre-multiply $B$ by a matrix with 3 columns.*False*$C$ is a $3\times 3$ matrix hence we can only post-multiply $C$ by a matrix with 3 rows and pre-multiply $C$ by a matrix with 3 columns*True**False*$C$ is a $3\times 3$ matrix hence we can only post-multiply $C$ by a matrix with 3 rows and pre-multiply $C$ by a matrix with 3 columns*True**True**True**False*$C$ is a $3\times 3$ matrix hence we can only post-multiply $C$ by a matrix with 3 rows and pre-multiply $C$ by a matrix with 3 columns*True**True*

If $A$ is a
$2\times 3$ matrix,
and $B$ is a
$3\times 4$ matrix, how many
columns does $AB$
have? (Enter your answer into the answer box.)

*Correct!*

$AB$ is a
$2\times 4$
matrix, and so has 4 columns.

*Incorrect.*

*Please try again.*

Remember that an $n\times m$
matrix, multiplied by an $m\times p$
matrix, is an $n\times p$
matrix.

Let $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill -3\hfill & \hfill 4\hfill \\ \hfill 2\hfill & \hfill 10\hfill & \hfill -7\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$
and $B=\left[\begin{array}{cc}\hfill 5\hfill & \hfill 0\hfill \\ \hfill -2\hfill & \hfill 11\hfill \\ \hfill -3\hfill & \hfill 12\hfill \end{array}\right]$.
Which of the following matrices is equal to the matrix
$AB$?
Exactly one option must be correct)

*Choice (a) is incorrect*

Since $A$ is
a $4\times 3$ matrix
and $B$ is a
$3\times 2$ matrix the
product $AB$
is a $4\times 2$
matrix.

*Choice (b) is incorrect*

The entry in row
1 and column 2 of $AB$
is $15$.

*Choice (c) is correct!*

*Choice (d) is incorrect*

The entry in row
4 and column 1 of $AB$
is $0$.

*Choice (e) is incorrect*

Since $A$ is
a $4\times 3$ matrix
and $B$ is a
$3\times 2$ matrix the
product $AB$ is a
$4\times 2$ matrix. In particular,
the product of $A$
and $B$ is
defined.

Let

For example, choice (a) should be True.

For example, choice (b) should be True.

For example, choice (c) should be False.

For example, choice (d) should be True.

For example, choice (e) should be False.

$$W=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 3\hfill & \hfill 2\hfill & \hfill 1\hfill \end{array}\right],\phantom{\rule{1em}{0ex}}X=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 2\hfill \end{array}\right]\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}Y=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill -2\hfill & \hfill 5\hfill \end{array}\right]$$

Which of the following statements are correct?
(Zero or more options can be correct)
*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 True.

*There is at least one mistake.*

For example, choice (c) should be False.

$XW$ is a
$3\times 3$ matrix so
the sum $XW+Y$
is does not make sense because we can only add matrices of the same size.

*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.

$WX=\left[\begin{array}{cc}\hfill 5\hfill & \hfill 4\hfill \\ \hfill 7\hfill & \hfill 0\hfill \end{array}\right]$so
$WXY=\left[\begin{array}{cc}\hfill -8\hfill & \hfill 25\hfill \\ \hfill 0\hfill & \hfill 7\hfill \end{array}\right]$.

*Correct!*

*True**True**False*$XW$ is a $3\times 3$ matrix so the sum $XW+Y$ is does not make sense because we can only add matrices of the same size.*True**False*$WX=\left[\begin{array}{cc}\hfill 5\hfill & \hfill 4\hfill \\ \hfill 7\hfill & \hfill 0\hfill \end{array}\right]$so $WXY=\left[\begin{array}{cc}\hfill -8\hfill & \hfill 25\hfill \\ \hfill 0\hfill & \hfill 7\hfill \end{array}\right]$.

Let

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 False.

For example, choice (f) should be True.

For example, choice (g) should be True.

$$W=\left[\begin{array}{cc}\hfill 1\hfill & \hfill -3\hfill \\ \hfill 2\hfill & \hfill -2\hfill \\ \hfill 3\hfill & \hfill 1\hfill \end{array}\right],X=\left[\begin{array}{ccc}\hfill 3\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 2\hfill & \hfill -1\hfill & \hfill 1\hfill \end{array}\right],\phantom{\rule{1em}{0ex}}Y=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill 2\hfill \end{array}\right]\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}Z=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 3\hfill \end{array}\right].$$

Which of the following statements are true ?
(Zero or more options can be correct)
*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.

$YW-{Z}^{2}=\left[\begin{array}{cc}\hfill 3\hfill & \hfill -10\hfill \\ \hfill -2\hfill & \hfill -7\hfill \end{array}\right].$

*There is at least one mistake.*

For example, choice (c) should be False.

$YW$ is a
$2\times 2$ matrix so its square
is defined. In fact, ${\left(YW\right)}^{2}=\left[\begin{array}{cc}\hfill -3\hfill & \hfill -63\hfill \\ \hfill 36\hfill & \hfill -12\hfill \end{array}\right]$

*There is at least one mistake.*

For example, choice (d) should be False.

If
$A$ is a matrix
then ${A}^{2}$ is defined
if and only if $A$
is a square matrix. Consequently, we cannot compute the square of either of the two
matrices $Y$
and $W$.

*There is at least one mistake.*

For example, choice (e) should be False.

$YW-{Z}^{2}=\left[\begin{array}{cc}\hfill 3\hfill & \hfill -10\hfill \\ \hfill -2\hfill & \hfill -7\hfill \end{array}\right].$

*There is at least one mistake.*

For example, choice (f) should be True.

*There is at least one mistake.*

For example, choice (g) should be True.

*Correct!*

*True**False*$YW-{Z}^{2}=\left[\begin{array}{cc}\hfill 3\hfill & \hfill -10\hfill \\ \hfill -2\hfill & \hfill -7\hfill \end{array}\right].$*False*$YW$ is a $2\times 2$ matrix so its square is defined. In fact, ${\left(YW\right)}^{2}=\left[\begin{array}{cc}\hfill -3\hfill & \hfill -63\hfill \\ \hfill 36\hfill & \hfill -12\hfill \end{array}\right]$*False*If $A$ is a matrix then ${A}^{2}$ is defined if and only if $A$ is a square matrix. Consequently, we cannot compute the square of either of the two matrices $Y$ and $W$.*False*$YW-{Z}^{2}=\left[\begin{array}{cc}\hfill 3\hfill & \hfill -10\hfill \\ \hfill -2\hfill & \hfill -7\hfill \end{array}\right].$*True**True*