## MATH1002 Quizzes

Quiz 7: Sums and products of matrices
Question 1 Questions
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)
 a) $2×3$ b) $3×2$ c) $3×4$ d) $4×3$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
$A$ has 3 rows and 4 columns and hence it is a $3×4$ matrix.
Choice (d) is incorrect
What is entry $\left(2,3\right)$ 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)
 a) 3 b) 4 c) 12 d) 14

Choice (a) is incorrect
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
Choice (d) is incorrect
Which sums can be made from the following matrices ?
$A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 4\hfill & \hfill 5\hfill & \hfill 6\hfill \end{array}\right],B=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 4\hfill \\ \hfill 5\hfill & \hfill 6\hfill \end{array}\right],C=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 4\hfill & \hfill 5\hfill & \hfill 6\hfill \\ \hfill 7\hfill & \hfill 8\hfill & \hfill 9\hfill \end{array}\right],D=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 4\hfill \end{array}\right].$
(Zero or more options can be correct)
 a) $A+A$ b) $A+B$ c) $A+C$ d) $A+D$ e) $B+A$ f) $B+B$ g) $B+D$ h) $C+A$ i) $C+C$ j) $C+D$ k) $D+A$ l) $D+C$ m) $D+D$ n) $D+B$

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 False.
Only matrices of the same size can be added.
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 False.
Only matrices of the same size can be added.
Correct!
1. True
2. False Only matrices of the same size can be added.
3. False Only matrices of the same size can be added.
4. False Only matrices of the same size can be added.
5. False Only matrices of the same size can be added.
6. True
7. False Only matrices of the same size can be added.
8. False Only matrices of the same size can be added.
9. True
10. False Only matrices of the same size can be added.
11. False
12. False Only matrices of the same size can be added.
13. True
14. False Only matrices of the same size can be added.
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)
 a) $\left[\begin{array}{ccc}\hfill 6\hfill & \hfill -5\hfill & \hfill -1\hfill \\ \hfill 2\hfill & \hfill 21\hfill & \hfill 5\hfill \end{array}\right]$ b) $\left[\begin{array}{cccccc}\hfill 1\hfill & \hfill -3\hfill & \hfill 4\hfill & \hfill 5\hfill & \hfill -2\hfill & \hfill -3\hfill \\ \hfill 2\hfill & \hfill 10\hfill & \hfill -7\hfill & \hfill 0\hfill & \hfill 11\hfill & \hfill 12\hfill \end{array}\right]$ c) $\left[\begin{array}{ccc}\hfill 6\hfill & \hfill -5\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 21\hfill & \hfill 5\hfill \end{array}\right]$ d) $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill -3\hfill & \hfill 4\hfill \\ \hfill 2\hfill & \hfill 10\hfill & \hfill -7\hfill \\ \hfill 5\hfill & \hfill -2\hfill & \hfill -3\hfill \\ \hfill 5\hfill & \hfill -2\hfill & \hfill -3\hfill \end{array}\right]$ e) $\left[\begin{array}{ccc}\hfill 6\hfill & \hfill 5\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 21\hfill & \hfill 5\hfill \end{array}\right]$ f) $A+B$ is undefined

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.
What is the matrix $7Z$ if $Z=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 3\hfill \\ \hfill 2\hfill & \hfill -1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 1\hfill \end{array}\right]$? Exactly one option must be correct)
 a) $\left[\begin{array}{ccc}\hfill 7\hfill & \hfill 0\hfill & \hfill 21\hfill \\ \hfill 2\hfill & \hfill -1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 1\hfill \end{array}\right]$ b) $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 3\hfill \\ \hfill 14\hfill & \hfill -7\hfill & \hfill 14\hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 1\hfill \end{array}\right]$ c) $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 3\hfill \\ \hfill 2\hfill & \hfill -1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 14\hfill & \hfill 7\hfill \end{array}\right]$ d) $\left[\begin{array}{ccc}\hfill 7\hfill & \hfill 0\hfill & \hfill 21\hfill \\ \hfill 14\hfill & \hfill -7\hfill & \hfill 14\hfill \\ \hfill 0\hfill & \hfill 14\hfill & \hfill 7\hfill \end{array}\right]$

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
$A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 4\hfill & \hfill 5\hfill & \hfill 6\hfill \end{array}\right],B=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 4\hfill \\ \hfill 5\hfill & \hfill 6\hfill \end{array}\right],C=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 4\hfill & \hfill 5\hfill & \hfill 6\hfill \\ \hfill 7\hfill & \hfill 8\hfill & \hfill 9\hfill \end{array}\right]\text{and}\phantom{\rule{1em}{0ex}}D=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 4\hfill \end{array}\right].$
Which of the following products can be made from these matrices ? (Zero or more options can be correct)
 a) $AB$ b) $AC$ c) $AD$ d) ${A}^{2}$ e) $BA$ f) $BD$ g) ${B}^{2}$ h) $CA$ i) $CB$ j) $CD$ k) $BC$ l) ${C}^{2}$ m) $DA$ n) $DC$ o) ${D}^{2}$

There is at least one mistake.
For example, choice (a) should be True.
$A$ is a $2×3$ matrix hence we can 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 $2×3$ matrix hence we can only post-multiply $A$ by a matrix with 3 rows.
There is at least one mistake.
For example, choice (d) should be False.
Only square matrices can be squared.
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.
Only square matrices can be squared.
There is at least one mistake.
For example, choice (h) should be False.
$C$ is a $3×3$ matrix hence we can only post-multiply $C$ by a matrix with 3 rows.
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 $3×3$ matrix hence we can only post-multiply $C$ by a matrix with 3 rows.
There is at least one mistake.
For example, choice (k) should be False.
$B$ is a $3×2$ matrix hence we can only post-multiply $B$ by a matrix with 2 rows.
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 $3×3$ matrix hence we can only pre-multiply $C$ by a matrix with 3 columns
There is at least one mistake.
For example, choice (o) should be True.
Correct!
1. True $A$ is a $2×3$ matrix hence we can post-multiply $A$ by a matrix with 3 rows and pre-multiply $A$ by a matrix with 2 columns.
2. True
3. False $A$ is a $2×3$ matrix hence we can only post-multiply $A$ by a matrix with 3 rows.
4. False Only square matrices can be squared.
5. True
6. True
7. False Only square matrices can be squared.
8. False $C$ is a $3×3$ matrix hence we can only post-multiply $C$ by a matrix with 3 rows.
9. True
10. False $C$ is a $3×3$ matrix hence we can only post-multiply $C$ by a matrix with 3 rows.
11. False $B$ is a $3×2$ matrix hence we can only post-multiply $B$ by a matrix with 2 rows.
12. True
13. True
14. False $C$ is a $3×3$ matrix hence we can only pre-multiply $C$ by a matrix with 3 columns
15. True
Which of the following matrices is the identity matrix ${I}_{4}$ ? Exactly one option must be correct)
 a) $\left[\begin{array}{cc}\hfill 4\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 4\hfill \end{array}\right]$ b) $\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ c) $\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ d) $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$

Choice (a) is incorrect
This is the matrix $4{I}_{2}$.
Choice (b) is correct!
The ${I}_{n}$ matrix is an $n×n$ matrix with 1’s down the diagonal and zeros elsewhere.
Choice (c) is incorrect
This is the matrix ${I}_{2}$.
Choice (d) is incorrect
This is the matrix ${I}_{3}$.
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)
 a) $\left[\begin{array}{cc}\hfill -1\hfill & \hfill 15\hfill \\ \hfill 11\hfill & \hfill 26\hfill \\ \hfill -3\hfill & \hfill 12\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$ b) $\left[\begin{array}{cc}\hfill -1\hfill & \hfill 0\hfill \\ \hfill -4\hfill & \hfill 26\hfill \\ \hfill 0\hfill & \hfill 12\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$ c) $\left[\begin{array}{cc}\hfill -1\hfill & \hfill 15\hfill \\ \hfill 11\hfill & \hfill 26\hfill \\ \hfill -2\hfill & \hfill 11\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$ d) $\left[\begin{array}{cc}\hfill -1\hfill & \hfill 15\hfill \\ \hfill 11\hfill & \hfill 26\hfill \\ \hfill -2\hfill & \hfill 11\hfill \\ \hfill 6\hfill & \hfill 0\hfill \end{array}\right]$ e) $AB$ is undefined

Choice (a) is incorrect
Since $A$ is a $4×3$ matrix and $B$ is a $3×2$ matrix the product $AB$ is a $4×2$ 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 $4×3$ matrix and $B$ is a $3×2$ matrix the product $AB$ is a $4×2$ matrix. In particular, the product of $A$ and $B$ is defined.
Let
$W=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 3\hfill & \hfill 2\hfill & \hfill 1\hfill \end{array}\right],\phantom{\rule{1em}{0ex}}X=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 2\hfill \end{array}\right]\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}Y=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill -2\hfill & \hfill 5\hfill \end{array}\right]$
Which of the following statements are correct? (Zero or more options can be correct)
 a) $WX+Y=\left[\begin{array}{cc}\hfill 5\hfill & \hfill 5\hfill \\ \hfill 5\hfill & \hfill 5\hfill \end{array}\right]$ b) $XY=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill -3\hfill \\ \hfill -4\hfill & \hfill 10\hfill \end{array}\right]$ c) $XW+Y=\left[\begin{array}{cc}\hfill 2\hfill & \hfill 5\hfill \\ \hfill 3\hfill & \hfill 0\hfill \end{array}\right]$ d) $YW=\left[\begin{array}{ccc}\hfill 3\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill 13\hfill & \hfill 6\hfill & \hfill -1\hfill \end{array}\right]$ e) $WXY=\left[\begin{array}{cc}\hfill -8\hfill & \hfill 20\hfill \\ \hfill -8\hfill & \hfill 15\hfill \end{array}\right]$

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 $3×3$ 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.
Since $WX=\left[\begin{array}{cc}\hfill 5\hfill & \hfill 4\hfill \\ \hfill -7\hfill & \hfill 0\hfill \end{array}\right]$, $WXY=\left[\begin{array}{cc}\hfill -8\hfill & \hfill 25\hfill \\ \hfill 0\hfill & \hfill 7\hfill \end{array}\right]$.
Correct!
1. True
2. True
3. False $XW$ is a $3×3$ matrix so the sum $XW+Y$ is does not make sense because we can only add matrices of the same size.
4. True
5. False Since $WX=\left[\begin{array}{cc}\hfill 5\hfill & \hfill 4\hfill \\ \hfill -7\hfill & \hfill 0\hfill \end{array}\right]$, $WXY=\left[\begin{array}{cc}\hfill -8\hfill & \hfill 25\hfill \\ \hfill 0\hfill & \hfill 7\hfill \end{array}\right]$.
Let
$W=\left[\begin{array}{cc}\hfill 1\hfill & \hfill -3\hfill \\ \hfill 2\hfill & \hfill -2\hfill \\ \hfill 3\hfill & \hfill 1\hfill \end{array}\right],X=\left[\begin{array}{ccc}\hfill 3\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 2\hfill & \hfill -1\hfill & \hfill 1\hfill \end{array}\right],\phantom{\rule{1em}{0ex}}Y=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill 2\hfill \end{array}\right]\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}Z=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 3\hfill \end{array}\right].$
Which of the following statements are true ? (Zero or more options can be correct)
 a) $WY-X=\left[\begin{array}{ccc}\hfill -2\hfill & \hfill 3\hfill & \hfill -7\hfill \\ \hfill 3\hfill & \hfill 4\hfill & \hfill -7\hfill \\ \hfill 1\hfill & \hfill 6\hfill & \hfill 1\hfill \end{array}\right]$ b) $YW-{Z}^{2}=\left[\begin{array}{cc}\hfill 5\hfill & \hfill -9\hfill \\ \hfill 0\hfill & \hfill -2\hfill \end{array}\right]$ c) The product ${\left(YW\right)}^{2}$ is not defined. d) ${\left(YW\right)}^{2}={Y}^{2}{W}^{2}$ e) $YW-{Z}^{2}=\left[\begin{array}{cc}\hfill 3\hfill & \hfill -10\hfill \\ \hfill -2\hfill & \hfill -7\hfill \end{array}\right].$

There is at least one mistake.
For example, choice (a) should be True.
There is at least one mistake.
For example, choice (b) should be False.
Try again.
There is at least one mistake.
For example, choice (c) should be False.
$YW$ is a $2×2$ 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 the either of the two matrices $Y$ or $W$.
There is at least one mistake.
For example, choice (e) should be True.
Correct!
1. True
2. False Try again.
3. False $YW$ is a $2×2$ matrix so its square is defined. In fact, ${\left(YW\right)}^{2}=\left[\begin{array}{cc}\hfill -3\hfill & \hfill -63\hfill \\ \hfill 36\hfill & \hfill -12\hfill \end{array}\right]$
4. 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 the either of the two matrices $Y$ or $W$.
5. True