Linear Algebra Quiz: Cramers Rule

What this quiz covers

This quiz focuses on Cramers Rule, giving you a quick way to practice the rules, question types, and explanations that matter most for Linear Algebra.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. The system has a unique solution.
2. The system has no solution. (correct answer)
3. The system has infinitely many solutions.
4. The variable $x_1$ must be zero, but no conclusion can be drawn about other variables.

Question 2

1. $y = \frac{\det \begin{pmatrix} 5 & -3 & 1 \\ -1 & 1 & -2 \\ 0 & 2 & 1 \end{pmatrix}}{\det \begin{pmatrix} 2 & -3 & 1 \\ 1 & 1 & -2 \\ -1 & 2 & 1 \end{pmatrix}}$
2. $y = \frac{\det \begin{pmatrix} 2 & 5 & 1 \\ 1 & -1 & -2 \\ -1 & 0 & 1 \end{pmatrix}}{\det \begin{pmatrix} 2 & -3 & 1 \\ 1 & 1 & -2 \\ -1 & 2 & 1 \end{pmatrix}}$ (correct answer)
3. $y = \frac{\det \begin{pmatrix} 2 & -3 & 5 \\ 1 & 1 & -1 \\ -1 & 2 & 0 \end{pmatrix}}{\det \begin{pmatrix} 2 & -3 & 1 \\ 1 & 1 & -2 \\ -1 & 2 & 1 \end{pmatrix}}$
4. $y = \frac{\det \begin{pmatrix} 2 & -3 & 1 \\ 1 & 1 & -2 \\ -1 & 2 & 1 \end{pmatrix}}{\det \begin{pmatrix} 2 & 5 & 1 \\ 1 & -1 & -2 \\ -1 & 0 & 1 \end{pmatrix}}$

Question 3

1. ($14/11$, $-12/11$) (correct answer)
2. ($-12/11$, $14/11$)
3. ($14$, $-12$)
4. ($11/14$, $-11/12$)

Question 4

1. The lines are parallel. (correct answer)
2. The lines are coincident.
3. The lines are perpendicular.
4. The lines intersect at the origin.

Question 5

1. The system has no solution.
2. The only solution is the trivial solution $\mathbf{x} = \mathbf{0}$.
3. The system has infinitely many solutions. (correct answer)
4. Cramer's rule is inapplicable, so no conclusion can be drawn from these results.

Question 6

1. 57
2. 31
3. -31
4. -57 (correct answer)

Question 7

1. Step 1
2. Step 2 (correct answer)
3. Step 3
4. The student made no mistake.

Question 8

1. $\det(A) \neq 0$, which implies $k \neq 14$.
2. $\det(A) = 0$, which implies $k = 18$.
3. $\det(A) = 0$, which implies $k = 14$. (correct answer)
4. The system must be inconsistent, which implies $k = -22$.

Question 9

1. $x_i' = kx_i$ (correct answer)
2. $x_i' = \frac{1}{k} x_i$
3. $x_i' = k^n x_i$, where $n$ is the dimension of the matrix $A$
4. $x_i' = x_i + k$

Question 10

1. 0
2. -30
3. 30 (correct answer)
4. 108

Question 11

1. The calculation of $D_x$ is incorrect; it should equal 8, making the system inconsistent rather than having $x = 0$
2. While both determinants are correctly calculated as zero, $\frac{0}{0}$ is indeterminate, not zero, and Cramer's rule doesn't apply here (correct answer)
3. The student should recognize that $D = 0$ means infinite solutions exist, so $x$ can be any value satisfying the original equation
4. The error is assuming the system has solutions; since $D = 0$, the system is inconsistent and has no solutions at all

Question 12

1. The new value of $z$ becomes $\frac{-72}{24} = -3$, exactly half the original value of $z = -6$
2. The new value of $z$ becomes $\frac{-36}{24} = -\frac{3}{2}$, which is half the original value of $z = -3$
3. The new value of $z$ remains $\frac{-36}{24} = -\frac{3}{2}$, unchanged from the original value despite coefficient changes (correct answer)
4. The new value of $z$ becomes $\frac{-72}{12} = -6$, which is double the original value of $z = -3$

Question 13

1. The incorrect value is $y = 2$, which is the negative of the correct value $y = -2$ (correct answer)
2. The incorrect value is $y = \frac{1}{2}$, which is the reciprocal of the correct value $y = 2$
3. The incorrect value is $y = 4$, which is twice the magnitude of the correct value $y = -2$
4. The incorrect value is $y = -4$, which has twice the magnitude of the correct value $y = -2$

Question 14

1. Yes, but only because this particular system has the special property that $D_x = 2D$, which is not generally true
2. No, swapping equations should not affect $D_x$ since it involves replacing columns, not rows, so the calculation is wrong
3. No, while $D$ changes sign when equations are swapped, $D_x$ should remain the same, giving $x = \frac{30}{-15} = -2$
4. Yes, swapping equations changes signs of both numerator and denominator equally, leaving $x = \frac{30}{15} = \frac{-30}{-15} = 2$ unchanged (correct answer)

Question 15

1. The solution cannot be determined without knowing the actual value of $D$, only the ratios between variables
2. The solution is $(x, y, z) = (\frac{1}{2}, -\frac{1}{2}, \frac{1}{4})$ obtained by careful application of Cramer's rule ratios
3. The relationships given are impossible since $D_x + D_y + D_z$ must equal $D$ for any valid system
4. The solution is $(x, y, z) = (2, -1, \frac{1}{2})$ and represents the unique solution to the system (correct answer)

Question 16

1. 1
2. $1/2$
3. 3
4. 2 (correct answer)

Question 17

1. -6
2. 12
3. $2/3$
4. 6 (correct answer)

Question 18

1. Cramer's rule can be applied, but will yield infinite solutions for each variable when the system is consistent
2. Cramer's rule cannot be applied because division by zero would occur, regardless of system consistency (correct answer)
3. Cramer's rule can be applied only if all numerator determinants are also zero, indicating infinite solutions
4. Cramer's rule yields zero for all variables, which represents the trivial solution when it exists