## MATH1002 Quizzes

Quiz 9: Inverses and elementary row operations
Question 1 Questions
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) $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 3\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill \hfill \end{array}\right]$ b) $\left[\begin{array}{ccc}\hfill 10\hfill & \hfill 20\hfill & \hfill 3\hfill \\ \hfill -3\hfill & \hfill -6\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 3\hfill & \hfill 1\hfill \end{array}\right]$ c) $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 2\hfill & \hfill 3\hfill & \hfill 1\hfill \end{array}\right]$ d) $\left[\begin{array}{ccc}\hfill -2\hfill & \hfill -1\hfill & \hfill -1\hfill \\ \hfill 3\hfill & \hfill 3\hfill & \hfill 2\hfill \\ \hfill 2\hfill & \hfill 3\hfill & \hfill 1\hfill \end{array}\right]$

Choice (a) is incorrect
Choice (b) is correct!
The two operations that need to be performed are ${R}_{2}:={R}_{2}-2{R}_{3}$ and ${R}_{1}:={R}_{1}-3{R}_{2}$.
Choice (c) is incorrect
You still need to perform two operations in order to reduce the left hand matrix to the identity matrix.
Choice (d) is incorrect
Suppose a sequence of elementary row operations has been performed on $\left[A|I\right]$ and has resulted in
$\left[\begin{array}{cccccc}\hfill 1\hfill & \hfill -\frac{1}{2}\hfill & \hfill 2\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right].$
Find ${A}^{-1}$. Exactly one option must be correct)
 a) $\left[\begin{array}{ccc}\hfill -\frac{5}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -1\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 2\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right]$ b) $\left[\begin{array}{ccc}\hfill \frac{3}{2}\hfill & \hfill 2\hfill & \hfill 2\hfill \\ \hfill -5\hfill & \hfill -2\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right]$ c) $\left[\begin{array}{ccc}\hfill -\frac{7}{2}\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill -1\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 2\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right]$ d) $\left[\begin{array}{ccc}\hfill \frac{1}{2}\hfill & \hfill 2\hfill & \hfill 4\hfill \\ \hfill 3\hfill & \hfill 2\hfill & \hfill 4\hfill \\ \hfill 2\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right]$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
The two elementary row operations that need to be performed are ${R}_{1}:={R}_{1}+\frac{1}{2}{R}_{2}$ and ${R}_{1}:={R}_{1}-2{R}_{3}$.
Choice (d) is incorrect
Suppose $A$ is a $9×9$ matrix that can be reduced to to the $9×9$ identity matrix by applying four elementary row operations. Let ${E}_{1}$, ${E}_{2}$, ${E}_{3}$, and ${E}_{4}$, respectively, be the elementary matrices corresponding to these four row operations. Which of the following is true ? Exactly one option must be correct)
 a) $A={E}_{1}{E}_{2}{E}_{3}{E}_{4}$ b) $A={E}_{4}{E}_{3}{E}_{2}{E}_{1}$ c) $A={{E}_{4}}^{-1}{{E}_{3}}^{-1}{{E}_{2}}^{-1}{{E}_{1}}^{-1}$ d) $A={{E}_{1}}^{-1}{{E}_{2}}^{-1}{{E}_{3}}^{-1}{{E}_{4}}^{-1}$ e) None of the above

Choice (a) is incorrect
Multiplying $A$ on the left by ${E}_{1}{E}_{2}{E}_{3}{E}_{4}$ would correspond to applying the fourth elementary row operation to $A$, followed by the third,m then second and then the first.
Choice (b) is incorrect
The matrix ${E}_{4}{E}_{3}{E}_{2}{E}_{1}$ is the inverse of $A$.
Choice (c) is incorrect
We have that ${E}_{4}{E}_{3}{E}_{2}{E}_{1}A=I$, where $I$ is the $9×9$ identity matrix.
Choice (d) is correct!
Recall that performing an elementary row operation is equivalent to multiplying on the left by the corresponding elementary matrix. So applying these four elementary operations to $A$ in order gives the sequence of matrices: $A\to {E}_{1}A\to {E}_{2}{E}_{1}A\to {E}_{3}{E}_{2}{E}_{1}A\to {E}_{4}{E}_{3}{E}_{2}{E}_{1}A$ Hence, ${E}_{4}{E}_{3}{E}_{2}{E}_{1}A=I$, where $I$ is the $9×9$ identity matrix. Multiplying on the left first by ${E}_{4}^{-1}$, and then by ${E}_{3}^{-1}$, ${E}_{2}^{-1}$ and ${E}_{1}^{-1}$ gives $A={{E}_{4}}^{-1}{{E}_{3}}^{-1}{{E}_{2}}^{-1}{{E}_{1}}^{-1}$ as claimed.
Choice (e) is incorrect
Recall that performing an elementary row operation is equivalent to multiplying on the left by the corresponding elementary matrix.
Suppose that a matrix $A$ can be transformed to ${I}_{4}$ by a sequence of elementary row operations and let ${E}_{1}$, ${E}_{2}$, ${E}_{3}$ and ${E}_{4}$ be the elementary matrices which correspond to these elementary row operations, in the order in which they are applied. Then $A={F}_{1}{F}_{2}{F}_{3}{F}_{4}$. If ${E}_{2}$ corresponds to the elementary row operation ${R}_{2}:={R}_{2}+3{R}_{3}$, find ${F}_{2}$. Exactly one option must be correct)
 a) $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ b) $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -3\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ c) $\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ d) $\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -3\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Since $A$ is transformed to ${I}_{4}$, $A$ must be a $4×4$ matrix and the elementary matrices are also $4×4$. Now
${E}_{2}=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{1em}{0ex}}and\phantom{\rule{1em}{0ex}}{F}_{2}={E}_{2}^{-1}=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -3\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$
.
Suppose the matrix $A$ can be transformed to ${I}_{3}$ by a sequence of elementary row operations and let ${E}_{1}$, ${E}_{2}$, ${E}_{3}$ and ${E}_{4}$ be the elementary matrices which correspond to these elementary row operations in the order in which they are applied. Then $A={F}_{1}{F}_{2}{F}_{3}{F}_{4}$. If ${F}_{4}$ corresponds to the elementary row operation ${R}_{1}:={R}_{1}+2{R}_{2}$, find ${E}_{4}$. Exactly one option must be correct)
 a) $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill -2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ b) $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ c) $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ d) $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$

Choice (a) is correct!
${F}_{4}={E}_{4}^{-1}$ so ${E}_{4}={F}_{4}^{-1}$.
${F}_{4}=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$hence ${E}_{4}=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill -2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Suppose the matrix $A$ can be transformed to ${I}_{2}$ by the sequence of elementary row operations ${R}_{2}:={R}_{2}-2{R}_{1}$, ${R}_{1}:={R}_{1}+3{R}_{2}$. Which of the following statements is true ? Exactly one option must be correct)
 a) $A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill -3\hfill \\ \hfill 2\hfill & \hfill -5\hfill \end{array}\right]$ b) ${A}^{-1}=\left[\begin{array}{cc}\hfill 1\hfill & \hfill -3\hfill \\ \hfill 2\hfill & \hfill -5\hfill \end{array}\right]$ c) $A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 3\hfill \\ \hfill -2\hfill & \hfill -5\hfill \end{array}\right]$ d) ${A}^{-1}=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 3\hfill \\ \hfill -2\hfill & \hfill -5\hfill \end{array}\right]$

Choice (a) is correct!
${E}_{1}=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill -2\hfill & \hfill 1\hfill \end{array}\right]$, ${E}_{2}=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$.
$A={E}_{1}^{-1}{E}_{2}^{-1}=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 1\hfill \end{array}\right]\left[\begin{array}{cc}\hfill 1\hfill & \hfill -3\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill 1\hfill & \hfill -3\hfill \\ \hfill 2\hfill & \hfill -5\hfill \end{array}\right]$.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
A certain matrix $A$ can be transformed to the $2×2$ identity matrix ${I}_{2}$ by the following sequence of elementary row operations:
${R}_{1}:=\frac{1}{2}{R}_{1},{R}_{2}:={R}_{2}-{R}_{1},{R}_{2}=\frac{1}{4}{R}_{2},{R}_{1}:={R}_{1}-2{R}_{2}.$
Find the matrices $A$ and ${A}^{-1}$. (Zero or more options can be correct)
 a) $A=\left[\begin{array}{ccc}\hfill \frac{3}{4}\hfill & \hfill -\frac{1}{2}\hfill & \hfill 0\hfill \\ \hfill -\frac{1}{8}\hfill & \hfill \frac{1}{4}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ b) ${A}^{-1}=\left[\begin{array}{ccc}\hfill 2\hfill & \hfill 4\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 6\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ c) $A=\left[\begin{array}{ccc}\hfill 2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -1\hfill & \hfill 6\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ d) ${A}^{-1}=\left[\begin{array}{ccc}\hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \frac{1}{12}\hfill & \hfill \frac{1}{6}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ e) $A=\left[\begin{array}{cc}\hfill 2\hfill & \hfill 4\hfill \\ \hfill 1\hfill & \hfill 6\hfill \end{array}\right]$ f) ${A}^{-1}=\left[\begin{array}{cc}\hfill \frac{3}{4}\hfill & \hfill -\frac{1}{2}\hfill \\ \hfill -\frac{1}{8}\hfill & \hfill \frac{1}{4}\hfill \end{array}\right]$ g) $A=\left[\begin{array}{cc}\hfill \frac{1}{2}\hfill & \hfill 0\hfill \\ \hfill \frac{1}{12}\hfill & \hfill \frac{1}{6}\hfill \end{array}\right]$ h) ${A}^{-1}=\left[\begin{array}{cc}\hfill 2\hfill & \hfill 0\hfill \\ \hfill -1\hfill & \hfill 6\hfill \end{array}\right]$

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 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.
${E}_{1}=\left[\begin{array}{cc}\hfill \frac{1}{2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$, ${E}_{2}=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill -1\hfill & \hfill 1\hfill \end{array}\right]$, ${E}_{3}=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{4}\hfill \end{array}\right]$, ${E}_{4}=\left[\begin{array}{cc}\hfill 1\hfill & \hfill -2\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$. So $A={E}_{1}^{-1}{E}_{2}^{-1}{E}_{3}^{-1}{E}_{4}^{-1}=\left[\begin{array}{cc}\hfill 2\hfill & \hfill 4\hfill \\ \hfill 1\hfill & \hfill 6\hfill \end{array}\right]$and ${A}^{-1}={E}_{4}{E}_{3}{E}_{2}{E}_{1}=\left[\begin{array}{cc}\hfill \frac{3}{4}\hfill & \hfill -\frac{1}{2}\hfill \\ \hfill -\frac{1}{8}\hfill & \hfill \frac{1}{4}\hfill \end{array}\right]$.
There is at least one mistake.
For example, choice (g) should be False.
There is at least one mistake.
For example, choice (h) should be False.
Correct!
1. False
2. False
3. False
4. False
5. True
6. True ${E}_{1}=\left[\begin{array}{cc}\hfill \frac{1}{2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$, ${E}_{2}=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill -1\hfill & \hfill 1\hfill \end{array}\right]$, ${E}_{3}=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{4}\hfill \end{array}\right]$, ${E}_{4}=\left[\begin{array}{cc}\hfill 1\hfill & \hfill -2\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$. So $A={E}_{1}^{-1}{E}_{2}^{-1}{E}_{3}^{-1}{E}_{4}^{-1}=\left[\begin{array}{cc}\hfill 2\hfill & \hfill 4\hfill \\ \hfill 1\hfill & \hfill 6\hfill \end{array}\right]$and ${A}^{-1}={E}_{4}{E}_{3}{E}_{2}{E}_{1}=\left[\begin{array}{cc}\hfill \frac{3}{4}\hfill & \hfill -\frac{1}{2}\hfill \\ \hfill -\frac{1}{8}\hfill & \hfill \frac{1}{4}\hfill \end{array}\right]$.
7. False
8. False
A matrix $A$ can be transformed to ${I}_{3}$, the $3×3$ identity matrixm by the following sequence of elementary row operations: $\begin{array}{llllllll}\hfill {R}_{2}& :={R}_{2}+2{R}_{1}\phantom{\rule{2em}{0ex}}& \hfill {R}_{3}& :={R}_{3}-{R}_{1}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {R}_{3}& :={R}_{3}+\frac{1}{2}{R}_{2}\phantom{\rule{2em}{0ex}}& \hfill {R}_{1}& :={R}_{1}-3{R}_{3}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {R}_{2}& :={R}_{2}+\frac{2}{3}{R}_{3}\phantom{\rule{2em}{0ex}}& \hfill {R}_{1}& :={R}_{1}+{R}_{2}.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$ Which of the following matrices is equal to $A$? Exactly one option must be correct)
 a) $A=\left[\begin{array}{ccc}\hfill \frac{31}{3}\hfill & \hfill -\frac{17}{6}\hfill & \hfill \frac{11}{3}\hfill \\ \hfill -\frac{10}{3}\hfill & \hfill \frac{4}{3}\hfill & \hfill -\frac{2}{3}\hfill \\ \hfill 2\hfill & \hfill -\frac{1}{2}\hfill & \hfill 1\hfill \end{array}\right]$ b) $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill -1\hfill & \hfill -3\hfill \\ \hfill 2\hfill & \hfill 3\hfill & \hfill -\frac{16}{3}\hfill \\ \hfill -1\hfill & \hfill -1\hfill & \hfill 4\hfill \end{array}\right]$ c) $A=\left[\begin{array}{ccc}\hfill 3\hfill & \hfill \frac{5}{6}\hfill & \hfill -\frac{7}{3}\hfill \\ \hfill 2\hfill & \hfill \frac{4}{3}\hfill & \hfill \frac{2}{3}\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 1\hfill \end{array}\right]$ d) $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill -1\hfill & \hfill 3\hfill \\ \hfill -2\hfill & \hfill 3\hfill & \hfill -\frac{20}{3}\hfill \\ \hfill 1\hfill & \hfill -\frac{3}{2}\hfill & \hfill \frac{13}{3}\hfill \end{array}\right]$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
The elementary matrices corresponding to the above elementary row operations are, in the order of application,
${E}_{1}=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$, ${E}_{2}=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill -1\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$, ${E}_{3}=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 1\hfill \end{array}\right]$,
${E}_{4}=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill -3\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$, ${E}_{5}=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill \frac{2}{3}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{1em}{0ex}}and\phantom{\rule{1em}{0ex}}{E}_{6}=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$.
$A={E}_{1}^{-1}{E}_{2}^{-1}{E}_{3}^{-1}{E}_{4}^{-1}{E}_{5}^{-1}{E}_{6}^{-1}=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill -1\hfill & \hfill 3\hfill \\ \hfill -2\hfill & \hfill 3\hfill & \hfill -\frac{20}{3}\hfill \\ \hfill 1\hfill & \hfill -\frac{3}{2}\hfill & \hfill \frac{13}{3}\hfill \end{array}\right]$.
Suppose that $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 3\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 3\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -2\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -3\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right].$ Find ${A}^{-1}$. Exactly one option must be correct)
 a) ${A}^{-1}=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{3}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill -2\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ b) ${A}^{-1}=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -3\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill -2\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ c) ${A}^{-1}=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{3}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill -3\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ d) ${A}^{-1}=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -3\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill -3\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
If $A={E}_{1}{E}_{2}\dots {E}_{5}$ then ${A}^{-1}={E}_{5}^{-1}{E}_{4}^{-1}\dots {E}_{1}^{-1}$
Choice (d) is incorrect
Suppose that ${A}^{-1}=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right].$ Find $A$. Exactly one option must be correct)
 a) $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ b) $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ c) $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ d) $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$

Choice (a) is incorrect
Choice (b) is correct!
If ${A}^{-1}={E}_{1}{E}_{2}\dots {E}_{5}$ then $A={E}_{5}^{-1}{E}_{4}^{-1}\dots {E}_{1}^{-1}$
Choice (c) is incorrect
Choice (d) is incorrect