## MATH1014 Quizzes

Quiz 11: Determinants
Question 1 Questions
Find the determinant of the matrix $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill -3\hfill & \hfill -4\hfill \\ \hfill 2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 3\hfill & \hfill -2\hfill & \hfill 5\hfill \end{array}\right]$. (Enter your answer into the answer box.)

Correct!

Expanding by the second row (for example),

$det\left(A\right)=-2\left|\begin{array}{cc}\hfill -3\hfill & \hfill -4\hfill \\ \hfill -2\hfill & \hfill 5\hfill \end{array}\right|+\left|\begin{array}{cc}\hfill 1\hfill & \hfill -4\hfill \\ \hfill 3\hfill & \hfill 5\hfill \end{array}\right|.$

Find the determinant of the matrix $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 7\hfill & \hfill 6\hfill \\ \hfill 5\hfill & \hfill 1\hfill & \hfill -2\hfill \\ \hfill 3\hfill & \hfill 8\hfill & \hfill 4\hfill \end{array}\right]$. (Enter your answer into the answer box.)

Correct!

Expanding by the first column (for example),

$det\left(A\right)=\left|\begin{array}{cc}\hfill 1\hfill & \hfill -2\hfill \\ \hfill 8\hfill & \hfill 4\hfill \end{array}\right|-5\left|\begin{array}{cc}\hfill 7\hfill & \hfill 6\hfill \\ \hfill 8\hfill & \hfill 4\hfill \end{array}\right|+3\left|\begin{array}{cc}\hfill 7\hfill & \hfill 6\hfill \\ \hfill 1\hfill & \hfill -2\hfill \end{array}\right|.$

Let $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill \end{array}\right]$. Which of the following statements is correct ? Exactly one option must be correct)
 a) $A$ is invertible since det $A=0$ b) $A$ is not invertible since det $A=0$ c) $A$ is invertible since det $A\ne 0$ d) $A$ is not invertible since det $A\ne 0$

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
Evaluate $\left|\begin{array}{ccc}\hfill 1\hfill & \hfill -3\hfill & \hfill -4\hfill \\ \hfill 2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill -6\hfill & \hfill -8\hfill \end{array}\right|$. (Enter your answer into the answer box.)

Correct!
Since $R3=2×R1$ the determinant is zero.
Note that $R3=2×R1$.
Suppose that $A$ is a $3×3$ matrix and det $\left(A\right)=-3$. What is det $\left(4A\right)$? Exactly one option must be correct)
 a) $-\frac{3}{4}$ b) $-3$ c) $-12$ d) $-192$

Choice (a) is incorrect
$4A$ is obtained by multiplying every entry in $A$ by 4. Remember that if one row (or column) of $A$ is multiplied by 4 then the determinant of the resulting matrix is equal to 4 times det $\left(A\right)$.
Choice (b) is incorrect
$4A$ is obtained by multiplying every entry in $A$ by 4. Remember that if one row (or column) of $A$ is multiplied by 4 then the determinant of the resulting matrix is equal to 4 times det $\left(A\right)$.
Choice (c) is incorrect
$4A$ is obtained by multiplying every entry in $A$ by 4. Remember that if one row (or column) of $A$ is multiplied by 4 then the determinant of the resulting matrix is equal to 4 times det $\left(A\right)$.
Choice (d) is correct!
Evaluate det $\left(A\right)$ if $A=\left[\begin{array}{ccc}\hfill 5\hfill & \hfill -3\hfill & \hfill -4\hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 6\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 9\hfill \end{array}\right]$. (Enter your answer into the answer box.)

Correct!
The product of the diagonal entries $=90$.
Since $A$ is triangular, the determinant is the product of the diagonal entries.
Let $A$ be a $3×3$ matrix, and suppose that the matrix $B$ is obtained from $A$ by the following row operations:

$R1↔R2;\phantom{\rule{1em}{0ex}}R3\to R3+5R2$ If det $\left(A\right)=12$, what is det $\left(B\right)$? Exactly one option must be correct)
 a) $-60$ b) $-12$ c) $12$ d) $60$

Choice (a) is incorrect
Adding a multiple of a row to another row does not change the value of the determinant.
Choice (b) is correct!
Choice (c) is incorrect
Swapping rows changes the sign of the determinant.
Choice (d) is incorrect
Adding a multiple of a row to another row does not change the value of the determinant.
If $A=\left[\begin{array}{ccc}\hfill 3\hfill & \hfill -1\hfill & \hfill 9\hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 6\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 5\hfill \end{array}\right]$, what is det $\left(7A\right)$? Exactly one option must be correct)
 a) $30$ b) $210$ c) $1470$ d) $10290$

Choice (a) is incorrect
det $\left(7A\right)={7}^{3}$ det $\left(A\right)$.
Choice (b) is incorrect
det $\left(7A\right)={7}^{3}$ det $\left(A\right)$.
Choice (c) is incorrect
det $\left(7A\right)={7}^{3}$ det $\left(A\right)$.
Choice (d) is correct!
Let $A=\left[\begin{array}{ccc}\hfill k\hfill & \hfill k\hfill & \hfill 4\hfill \\ \hfill 0\hfill & \hfill k\hfill & \hfill 5\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill k\hfill \end{array}\right]$. Which one of the following statements is true? Exactly one option must be correct)
 a) If $k=0$ then $A$ is invertible. b) $A$ is invertible if $k$ is any real number other than $0$ or $4$. c) $A$ is invertible if $k=0$ or $4$. d) $A$ is invertible if $k$ is any real number other than $0$, $2$ or $-2$. e) $A$ is invertible if $k=0$, $2$ or $-2$.

Choice (a) is incorrect
$A$ is invertible if det $\left(A\right)\ne 0$.
Choice (b) is incorrect
Choice (c) is incorrect
$A$ is invertible if det $\left(A\right)\ne 0$.
Choice (d) is correct!
Choice (e) is incorrect
$A$ is invertible if det $\left(A\right)\ne 0$.
Let $A=\left[\begin{array}{ccc}\hfill -2\hfill & \hfill 1\hfill & \hfill 4\hfill \\ \hfill 0\hfill & \hfill 3\hfill & \hfill 7\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$. Find any values of $\lambda$ such that det $\left(A-\lambda I\right)=0$. Exactly one option must be correct)
 a) $\lambda =-2,0$ b) $\lambda =0,1$ c) $\lambda =-2,1,4$ d) $\lambda =-2,1,3$ e) $\lambda =0,1,3$ f) $\lambda =1,4,7$ g) $\lambda =0,3,7$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Choice (e) is incorrect
Choice (f) is incorrect
Choice (g) is incorrect