## MATH1002 Quizzes

Quiz 5: Row echelon form
Question 1 Questions
Which of the following arrays are $3×2$ matrices? (More than one answer may be correct.) (Zero or more options can be correct)
 a) $\left[\begin{array}{ccc}\hfill 3\hfill & \hfill -2\hfill & \hfill -1\hfill \\ \hfill 10\hfill & \hfill 0\hfill & \hfill 22\hfill \end{array}\right]$ b) $\left[\begin{array}{cc}\hfill 3\hfill & \hfill 10\hfill \\ \hfill -2\hfill & \hfill 0\hfill \\ \hfill -1\hfill & \hfill 22\hfill \end{array}\right]$ c) $\left[\begin{array}{cc}\hfill 2\hfill & \hfill 5\hfill \\ \hfill 3\hfill & \hfill -9\hfill \\ \hfill 5\hfill \end{array}\right]$ d) $\left[\begin{array}{ccc}\hfill 3\hfill & \hfill -2\hfill & \hfill -1\hfill \\ \hfill \hfill & \hfill 10\hfill & \hfill -3\hfill \end{array}\right]$ e) $\left[\begin{array}{ccc}\hfill 3\hfill & \hfill -2\hfill & \hfill -1\hfill \\ \hfill 3\hfill & \hfill -2\hfill & \hfill -1\hfill \\ \hfill 4\hfill & \hfill 10\hfill & \hfill -3\hfill \end{array}\right]$

There is at least one mistake.
For example, choice (a) should be False.
This is a $2×3$ matrix (2 rows, 3 columns).
There is at least one mistake.
For example, choice (b) should be True.
This matrix has 3 rows and 2 columns.
There is at least one mistake.
For example, choice (c) should be False.
This is not a matrix (i.e., is not a rectangular array) since not all rows have the same numbers of entries (and not all columns have the same number of entries).
There is at least one mistake.
For example, choice (d) should be False.
This is not a matrix (i.e., is not a rectangular array) since not all rows have the same numbers of entries (and not all columns have the same number of entries).
There is at least one mistake.
For example, choice (e) should be False.
This is a $3×3$ matrix.
Correct!
1. False This is a $2×3$ matrix (2 rows, 3 columns).
2. True This matrix has 3 rows and 2 columns.
3. False This is not a matrix (i.e., is not a rectangular array) since not all rows have the same numbers of entries (and not all columns have the same number of entries).
4. False This is not a matrix (i.e., is not a rectangular array) since not all rows have the same numbers of entries (and not all columns have the same number of entries).
5. False This is a $3×3$ matrix.
Which of the following is the elementary row operation which transforms the matrix $A=\left[\begin{array}{cccc}\hfill 2\hfill & \hfill 4\hfill & \hfill 2\hfill & \hfill -1\hfill \\ \hfill 2\hfill & \hfill 5\hfill & \hfill 1\hfill & \hfill 6\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{2em}{0ex}}\text{into}\phantom{\rule{2em}{0ex}}B=\left[\begin{array}{cccc}\hfill 2\hfill & \hfill 4\hfill & \hfill 2\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -1\hfill & \hfill 7\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\phantom{\rule{1em}{0ex}}\text{?}$ Exactly one option must be correct)
 a) ${R}_{2}:={R}_{1}-2{R}_{2}$ b) ${R}_{1}:={R}_{1}-{R}_{2}$ c) ${R}_{1}:={R}_{2}-{R}_{1}$ d) ${R}_{1}↔{R}_{3}$ e) ${R}_{2}:={R}_{2}-{R}_{1}$

Choice (a) is incorrect
This is not an elementary row operation.
Choice (b) is incorrect
This will alter row 1 and leave row 2 unchanged. So applying this elementary row operation to $A$ will give the matrix $\left[\begin{array}{cccc}\hfill 0\hfill & \hfill -1\hfill & \hfill -1\hfill & \hfill -7\hfill \\ \hfill 2\hfill & \hfill 5\hfill & \hfill 1\hfill & \hfill 6\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ give $B$ from $A$.
Choice (c) is incorrect
This is not an elementary row operation.
Choice (d) is incorrect
This operation interchanges row 1 and row 3. Applying this operation to $A$ gives the matrix $\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 5\hfill & \hfill 1\hfill & \hfill 6\hfill \\ \hfill 2\hfill & \hfill 4\hfill & \hfill 2\hfill & \hfill -1\hfill \end{array}\right]$
Choice (e) is correct!
Which of the following matrices is the augmented matrix for the following system of equations: $\begin{array}{llll}\hfill w+2x-5& =0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 2w+3x+6z+1& =0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 3x-7-2z& =0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}& \hfill \end{array}$ in the four unknowns $w,x,y$ and $z$? Exactly one option must be correct)
 a) $\left[\begin{array}{cccc}\hfill 1& \hfill 2& \hfill 0& \hfill 5\\ \hfill 2& \hfill 3& \hfill 6& \hfill -1\\ \hfill 0& \hfill 3& \hfill -2& \hfill 7\end{array}\right]$ b) $\left[\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 2& \hfill -5\\ \hfill 2& \hfill 3& \hfill 6& \hfill 1\\ \hfill 0& \hfill 3& \hfill -7& \hfill -2\end{array}\right]$ c) $\left[\begin{array}{ccccc}\hfill 1& \hfill 2& \hfill 0& \hfill 0& \hfill 5\\ \hfill 2& \hfill 3& \hfill 0& \hfill 6& \hfill -1\\ \hfill 0& \hfill 3& \hfill 0& \hfill 2& \hfill 7\end{array}\right]$ d) $\left[\begin{array}{cccc}\hfill 1& \hfill 2& \hfill 0& \hfill 5\\ \hfill 2& \hfill 3& \hfill 6& \hfill 1\\ \hfill 0& \hfill 3& \hfill 2& \hfill 7\end{array}\right]$ e) $\left[\begin{array}{ccccc}\hfill 1& \hfill 2& \hfill 0& \hfill 0& \hfill 5\\ \hfill 2& \hfill 3& \hfill 0& \hfill 6& \hfill -1\\ \hfill 0& \hfill 3& \hfill 0& \hfill -2& \hfill 7\end{array}\right]$

Choice (a) is incorrect
The augmented matrix should have 5 columns — one for each of the 4 unknowns and one for the constants on the right hand side.
Choice (b) is incorrect
This is just the array of coefficients in the equations.
Choice (c) is incorrect
One of the coefficients in the fourth column is wrong.
Choice (d) is incorrect
Even though the variable $y$ does not appear in any of the equations it still must have a column in the augmented matrix.
Choice (e) is correct!
The system of equations can be rewritten with the constant terms on the right hand side, giving $\begin{array}{llll}\hfill w+2x& =5\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 2w+3x+6z& =-1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 3x-2z& =7\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}& \hfill \end{array}$
Which of the following matrices are in row echelon form? (More than one answer may be correct.) (Zero or more options can be correct)
 a) $\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 3\hfill & \hfill 0\hfill & \hfill 5\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$ b) $\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 3\hfill & \hfill 0\hfill & \hfill 5\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$ c) $\left[\begin{array}{cccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$ d) $\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 3\hfill & \hfill 6\hfill & \hfill 5\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$ e) $\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$

There is at least one mistake.
For example, choice (a) should be False.
The first non-zero entry in row 3 is not 1, so this is not in row echelon form.
There is at least one mistake.
For example, choice (b) should be True.
This is in row echelon form because the first non–zero entry in each non–zero row is equal to $1$, and each leading $1$ is in a later column of the matrix than the leadings $1$s in previous rows, with the zero rows occurring last.
There is at least one mistake.
For example, choice (c) should be False.
The leading $1$s in rows 1 and 2 appear in the same column.
There is at least one mistake.
For example, choice (d) should be True.
This is in row echelon form. Note, however, that this matrix is not in reduced row echelon form since the entry in row 1, column 3 is non–zero.
There is at least one mistake.
For example, choice (e) should be False.
The zero rows occur at the bottom of matrices which are in row echelon form.
Correct!
1. False The first non-zero entry in row 3 is not 1, so this is not in row echelon form.
2. True This is in row echelon form because the first non–zero entry in each non–zero row is equal to $1$, and each leading $1$ is in a later column of the matrix than the leadings $1$s in previous rows, with the zero rows occurring last.
3. False The leading $1$s in rows 1 and 2 appear in the same column.
4. True This is in row echelon form. Note, however, that this matrix is not in reduced row echelon form since the entry in row 1, column 3 is non–zero.
5. False The zero rows occur at the bottom of matrices which are in row echelon form.
Which of the following matrices are in row echelon form? (More than one answer may be correct.) (Zero or more options can be correct)
 a) $\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 5\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ b) $\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 5\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$ c) $\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 5\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$ d) $\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 2\hfill & \hfill -1\hfill & \hfill 6\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -1\hfill & \hfill 4\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 4\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 7\hfill \end{array}\right]$ e) $\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 2\hfill & \hfill -7\hfill & \hfill 6\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -1\hfill & \hfill 4\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 4\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$

There is at least one mistake.
For example, choice (a) should be False.
The (only) zero row is not the last row, so $A$ is not in row echelon form.
There is at least one mistake.
For example, choice (b) should be False.
The leading 1 in row 2 is in the same column as the leading $1$ is in row 1, so this matrix is not in row echelon form.
There is at least one mistake.
For example, choice (c) should be False.
The first non-zero in row 2 is not 1, so this matrix is not a row echelon matrix.
There is at least one mistake.
For example, choice (d) should be False.
The leading entries in rows 3 and 4 are not equal to 1 so this matrix is not in row echelon form.
There is at least one mistake.
For example, choice (e) should be True.
Any “triangular” matrix with $1$s down its diagonal is automatically in row echelon form.
Correct!
1. False The (only) zero row is not the last row, so $A$ is not in row echelon form.
2. False The leading 1 in row 2 is in the same column as the leading $1$ is in row 1, so this matrix is not in row echelon form.
3. False The first non-zero in row 2 is not 1, so this matrix is not a row echelon matrix.
4. False The leading entries in rows 3 and 4 are not equal to 1 so this matrix is not in row echelon form.
5. True Any “triangular” matrix with $1$s down its diagonal is automatically in row echelon form.
Which of the statements below correctly describes the solutions of the system of equations with the following augmented matrix?
$\left[\begin{array}{ccccc}\hfill 1& \hfill 1& \hfill -3& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1& \hfill 5& \hfill 1\\ \hfill 0& \hfill 0& \hfill 1& \hfill 6& \hfill 2\\ \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 2\end{array}\right]$
Exactly one option must be correct)
 a) The solution is unique. b) There are infinitely many solutions. c) The system is inconsistent, there are no solutions. d) There is not enough information to decide.

Choice (a) is incorrect
Hint: apply suitable elementary row operations to the augmented matrix.
Choice (b) is incorrect
Hint: apply suitable elementary row operations to the augmented matrix.
Choice (c) is correct!
Reduce the matrix to row echelon form.
$\begin{array}{ccc}\hfill \left[\begin{array}{ccccc}\hfill 1& \hfill 1& \hfill -3& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1& \hfill 5& \hfill 1\\ \hfill 0& \hfill 0& \hfill 1& \hfill 6& \hfill 2\\ \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 2\end{array}\right]\hfill & \hfill \underset{}{\overset{{R}_{3}:={R}_{3}-{R}_{2}}{\to }}\hfill & \hfill \left[\begin{array}{ccccc}\hfill 1& \hfill 1& \hfill -3& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1& \hfill 5& \hfill 1\\ \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 1\\ \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 2\end{array}\right]\hfill \\ & & \\ \hfill \hfill & \hfill \underset{}{\overset{{R}_{4}:={R}_{4}-{R}_{3}}{\to }}\hfill & \hfill \left[\begin{array}{ccccc}\hfill 1& \hfill 1& \hfill -3& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1& \hfill 5& \hfill 1\\ \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 1\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1\end{array}\right]\hfill \end{array}$

The last row indicates that the system is inconsistent and has no solution.
Choice (d) is incorrect
It is always possible to decide such questions by reducing the augmented matrix to row echelon form.
Which of the statements below correctly describes the solutions of the system of equations with the following augmented matrix?
$\left[\begin{array}{ccccc}\hfill 1& \hfill 2& \hfill 7& \hfill 1& \hfill 3\\ \hfill 0& \hfill 1& \hfill -5& \hfill 7& \hfill 1\\ \hfill 0& \hfill 1& \hfill -4& \hfill 10& \hfill 2\\ \hfill 0& \hfill 0& \hfill 1& \hfill 4& \hfill 2\end{array}\right]$
Exactly one option must be correct)
 a) The solution is unique. b) There are infinitely many solutions. c) The system is inconsistent, there are no solutions. d) There is not enough information to decide.

Choice (a) is correct!
Reduce the matrix to row echelon form.
$\begin{array}{ccc}\hfill \left[\begin{array}{ccccc}\hfill 1& \hfill 2& \hfill 7& \hfill 1& \hfill 3\\ \hfill 0& \hfill 1& \hfill -5& \hfill 7& \hfill 1\\ \hfill 0& \hfill 1& \hfill -4& \hfill 10& \hfill 2\\ \hfill 0& \hfill 0& \hfill 1& \hfill 4& \hfill 2\end{array}\right]\hfill & \hfill \underset{}{\overset{{R}_{3}:={R}_{3}-{R}_{2}}{\to }}\hfill & \hfill \left[\begin{array}{ccccc}\hfill 1& \hfill 2& \hfill 7& \hfill 1& \hfill 3\\ \hfill 0& \hfill 1& \hfill -5& \hfill 7& \hfill 1\\ \hfill 0& \hfill 0& \hfill 1& \hfill 3& \hfill 1\\ \hfill 0& \hfill 0& \hfill 1& \hfill 4& \hfill 2\end{array}\right]\hfill \\ & & \\ \hfill \hfill & \hfill \underset{}{\overset{{R}_{4}:={R}_{4}-{R}_{3}}{\to }}\hfill & \hfill \left[\begin{array}{ccccc}\hfill 1& \hfill 2& \hfill 7& \hfill 1& \hfill 3\\ \hfill 0& \hfill 1& \hfill -5& \hfill 7& \hfill 1\\ \hfill 0& \hfill 0& \hfill 1& \hfill 3& \hfill 1\\ \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 1\end{array}\right]\hfill \end{array}$

There is a leading one in each column of the coefficient matrix, so the system has a unique solution.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
It is always possible to decide such questions by reducing the augmented matrix to row echelon form.
Which of the following correctly describes the nature of the solution of the system of equations corresponding to the following augmented matrix?
$\left[\begin{array}{ccccc}\hfill 1& \hfill 0& \hfill 1& \hfill 2& \hfill 4\\ \hfill 0& \hfill 1& \hfill 3& \hfill -1& \hfill 2\\ \hfill 0& \hfill 1& \hfill 0& \hfill -4& \hfill -7\\ \hfill 0& \hfill 0& \hfill 1& \hfill 1& \hfill 3\end{array}\right]$
Exactly one option must be correct)
 a) The solution is unique. b) There are infinitely many solutions. c) The system is inconsistent, there are no solutions. d) There is not enough information to decide.

Choice (a) is incorrect
Choice (b) is correct!
Reduce the matrix to row echelon form.
$\begin{array}{ccc}\hfill \left[\begin{array}{ccccc}\hfill 1& \hfill 0& \hfill 1& \hfill 2& \hfill 4\\ \hfill 0& \hfill 1& \hfill 3& \hfill -1& \hfill 2\\ \hfill 0& \hfill 1& \hfill 0& \hfill -4& \hfill -7\\ \hfill 0& \hfill 0& \hfill 1& \hfill 1& \hfill 3\end{array}\right]\hfill & \hfill \underset{}{\overset{{R}_{3}:={R}_{3}-{R}_{2}}{\to }}\hfill & \hfill \left[\begin{array}{ccccc}\hfill 1& \hfill 0& \hfill 1& \hfill 2& \hfill 4\\ \hfill 0& \hfill 1& \hfill 3& \hfill -1& \hfill 2\\ \hfill 0& \hfill 0& \hfill -3& \hfill -3& \hfill -9\\ \hfill 0& \hfill 0& \hfill 1& \hfill 1& \hfill 3\end{array}\right]\hfill \\ & & \\ \hfill \hfill & \hfill \underset{}{\overset{{R}_{3}:={R}_{3}-\frac{1}{3}{R}_{2}}{\to }}\hfill & \hfill \left[\begin{array}{ccccc}\hfill 1& \hfill 0& \hfill 1& \hfill 2& \hfill 4\\ \hfill 0& \hfill 1& \hfill 3& \hfill -1& \hfill 2\\ \hfill 0& \hfill 0& \hfill 1& \hfill 1& \hfill 3\\ \hfill 0& \hfill 0& \hfill 1& \hfill 1& \hfill 3\end{array}\right]\hfill \\ & & \\ \hfill \hfill & \hfill \underset{}{\overset{{R}_{4}:={R}_{4}-{R}_{3}}{\to }}\hfill & \hfill \left[\begin{array}{ccccc}\hfill 1& \hfill 0& \hfill 1& \hfill 2& \hfill 4\\ \hfill 0& \hfill 1& \hfill 3& \hfill -1& \hfill 2\\ \hfill 0& \hfill 0& \hfill 1& \hfill 1& \hfill 3\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\end{array}\right]\hfill \end{array}$

The fourth column does not have a leading one, so the system has infinitely many solutions.
Choice (c) is incorrect
Choice (d) is incorrect
It is always possible to decide such questions by reducing the augmented matrix to row echelon form.
Which is the correct solution for the system of equations corresponding to the augmented matrix $\left[\begin{array}{cccc}\hfill 1& \hfill 2& \hfill 1& \hfill 1\\ \hfill 0& \hfill 1& \hfill 1& \hfill 2\\ \hfill 0& \hfill 0& \hfill 1& \hfill 4\end{array}\right]?$ Exactly one option must be correct)
 a) $x=3$, $y=-3$, $z=4$ b) $x=-1$, $y=-1$, $z=4$ c) $x=1$, $y=-2$, $z=4$ d) $x=-3$, $y=0$, $z=4$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Using back substitution we see that $\begin{array}{rcll}z& =& 4& \text{}\\ y& =& 2-z=2-4=-2& \text{}\\ x& =& -2y-z+1& \text{}\\ & =& 4-4+1=1.& \text{}\end{array}$
Choice (d) is incorrect
Which of the statements below correctly describe the relationship between the following matrices?
$\begin{array}{cc}\hfill A=\left[\begin{array}{ccc}\hfill 1& \hfill 0& \hfill 1\\ \hfill 0& \hfill 1& \hfill 1\end{array}\right],\hfill & \hfill B=\left[\begin{array}{ccc}\hfill 1& \hfill 0& \hfill 2\\ \hfill 0& \hfill 1& \hfill 1\end{array}\right],\hfill \\ & \\ \hfill C=\left[\begin{array}{ccc}\hfill 1& \hfill 1& \hfill 2\\ \hfill 0& \hfill 1& \hfill 1\end{array}\right],\hfill & \hfill D=\left[\begin{array}{ccc}\hfill 1& \hfill 2& \hfill 4\\ \hfill 0& \hfill 1& \hfill 1\end{array}\right].\hfill \end{array}$ (More than one answer may be correct.) (Zero or more options can be correct)
 a) $A$ is row equivalent to $B$ b) $A$ is row equivalent to $C$ c) $B$ is row equivalent to $D$. d) $A$ is row equivalent to $D$ e) $B$ is row equivalent to $C$.

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 True.
$A$ is row equivalent to $C$ because
$C=\left[\begin{array}{ccc}\hfill 1& \hfill 1& \hfill 2\\ \hfill 0& \hfill 1& \hfill 1\end{array}\right]\underset{}{\overset{{R}_{1}:={R}_{1}-{R}_{2}}{\to }}\left[\begin{array}{ccc}\hfill 1& \hfill 0& \hfill 1\\ \hfill 0& \hfill 1& \hfill 1\end{array}\right]=A$
There is at least one mistake.
For example, choice (c) should be True.
$B$ is row equivalent to $D$ because
$D=\left[\begin{array}{ccc}\hfill 1& \hfill 2& \hfill 4\\ \hfill 0& \hfill 1& \hfill 1\end{array}\right]\underset{}{\overset{{R}_{1}:={R}_{1}-2{R}_{2}}{\to }}\left[\begin{array}{ccc}\hfill 1& \hfill 0& \hfill 2\\ \hfill 0& \hfill 1& \hfill 1\end{array}\right]=B$
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 False.
Correct!
1. False
2. True $A$ is row equivalent to $C$ because
$C=\left[\begin{array}{ccc}\hfill 1& \hfill 1& \hfill 2\\ \hfill 0& \hfill 1& \hfill 1\end{array}\right]\underset{}{\overset{{R}_{1}:={R}_{1}-{R}_{2}}{\to }}\left[\begin{array}{ccc}\hfill 1& \hfill 0& \hfill 1\\ \hfill 0& \hfill 1& \hfill 1\end{array}\right]=A$
3. True $B$ is row equivalent to $D$ because
$D=\left[\begin{array}{ccc}\hfill 1& \hfill 2& \hfill 4\\ \hfill 0& \hfill 1& \hfill 1\end{array}\right]\underset{}{\overset{{R}_{1}:={R}_{1}-2{R}_{2}}{\to }}\left[\begin{array}{ccc}\hfill 1& \hfill 0& \hfill 2\\ \hfill 0& \hfill 1& \hfill 1\end{array}\right]=B$
4. False
5. False