Linear Algebra Quiz: Invertible Matrices

What this quiz covers

This quiz focuses on Invertible Matrices, 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. Both $A$ and $B$ must be invertible. (correct answer)
2. At least one of $A$ or $B$ must be invertible.
3. Neither $A$ nor $B$ is necessarily invertible.
4. $A$ must be the inverse of $B$.

Question 2

1. The three planes must intersect at a single, unique point.
2. The three planes may intersect along a common line or not at all. (correct answer)
3. The three planes must be parallel to each other.
4. The three planes must be coincident (i.e., they are all the same plane).

Question 3

1. It has a unique solution for every $b \in \mathbb{R}^n$. (correct answer)
2. It has a solution only when $b$ is the zero vector.
3. It has infinitely many solutions for every non-zero $b \in \mathbb{R}^n$.
4. It may have no solution for some vectors $b \in \mathbb{R}^n$.

Question 4

1. $k=1$
2. $k=2$
3. $k=3$ (correct answer)
4. Any value of $k$ except $3$

Question 5

1. $k = -6$ makes the system consistent for all $\mathbf{b}$
2. $k = 0$ makes the matrix invertible and ensures solutions exist
3. $k = 3$ maximizes the rank and guarantees invertibility
4. No value of $k$ makes the system solvable for every $\mathbf{b}$ (correct answer)

Question 6

1. Matrix $(B + I)$ is always invertible since $B$ is invertible
2. Matrix $(B + I)$ is not invertible because $-1$ is an eigenvalue of $B$ (correct answer)
3. Matrix $(B + I)$ may or may not be invertible depending on other eigenvalues
4. Matrix $(B + I)$ is invertible if and only if $\det(B + I) \neq 0$

Question 7

1. $M$ is invertible if and only if $C$ is invertible, regardless of $B$ (correct answer)
2. $M$ is invertible if and only if both $A$ and $C$ are invertible and $B = 0$
3. $M$ is invertible if and only if $\det(A) + \det(C) \neq 0$
4. $M$ is always invertible since $A$ is given to be invertible

Question 8

1. The systems have identical solution sets with $\mathbf{x} = \mathbf{y}$ for all solutions
2. The systems are equivalent with solutions related by $\mathbf{y} = P\mathbf{x}$
3. The second system always has more solutions than the first system
4. The systems have the same number of solutions with $\mathbf{x} = P\mathbf{y}$ relating them (correct answer)

Question 9

1. At most 5 linearly independent columns since $B$ is in $\mathbb{R}^{5 \times 3}$
2. At most 2 linearly independent columns since $\text{nullity}(A) = 2$
3. At most 3 linearly independent columns since $\text{rank}(A) = 3$ (correct answer)
4. Exactly 3 linearly independent columns for any consistent system

Question 10

1. $\text{rank}([A | \mathbf{b}]) = 3$ because the system is consistent
2. $\text{rank}([A | \mathbf{b}]) = 2$ since the system has infinitely many solutions (correct answer)
3. $\text{rank}([A | \mathbf{b}]) = 1$ because there are multiple free variables
4. $\text{rank}([A | \mathbf{b}])$ could be either 2 or 3 depending on $\mathbf{b}$

Question 11

1. Matrix $A$ is invertible and has rank 3
2. Matrix $A$ is not invertible and has rank less than 3
3. The given conditions are contradictory and cannot occur simultaneously (correct answer)
4. Matrix $A$ must be the zero matrix with rank 0

Question 12

1. $k \neq 2$ is the only condition needed for invertibility (correct answer)
2. $k \neq 0$ and $k \neq 4$ are required for invertibility
3. $k \neq 2$ and $k \neq 4$ ensure that $(A - 2I)$ is invertible
4. $k \neq -1$, $k \neq 2$, and $k \neq 3$ are needed for invertibility

Question 13

1. The system must have a unique solution.
2. The system must have infinitely many solutions.
3. The system has either no solution or infinitely many solutions. (correct answer)
4. The system must have no solution.

Question 14

1. The equation $Ax=b$ has a unique solution for any $b \in \mathbb{R}^n$.
2. The equation $Ax=b$ must be inconsistent for any non-zero $b \in \mathbb{R}^n$.
3. The equation $Ax=b$ has either no solution or infinitely many solutions. (correct answer)
4. The equation $Ax=0$ has only the trivial solution.

Question 15

1. $c=2$ only
2. $c=-2$ only
3. $c=\sqrt{5}$ or $c=-\sqrt{5}$
4. $c=2$ or $c=-2$ (correct answer)

Question 16

1. $x = \frac{b}{A}$
2. $x = bA^{-1}$ (correct answer)
3. $x = (A-I)b$
4. $x = b-A$

Question 17

1. The system $Ax=b$ has a unique solution for every $b \in \mathbb{R}^4$.
2. The determinant of $A$ is non-zero.
3. The columns of $A$ span $\mathbb{R}^4$.
4. The system $Ax=0$ has infinitely many solutions. (correct answer)

Question 18

1. The equation $Ax=b$ has a unique solution for any $b \in \mathbb{R}^n$. (correct answer)
2. The equation $Ax=b$ has infinitely many solutions for any $b \in \mathbb{R}^n$.
3. The equation $Ax=0$ has a non-trivial solution.
4. The determinant of $A$ is 0.

Question 19

1. The system is consistent.
2. The number of equations equals the number of unknowns.
3. The vector $b$ is not the zero vector.
4. The columns of the coefficient matrix $A$ are linearly independent. (correct answer)

Question 20

1. Matrix $A$ is always invertible because it satisfies a polynomial equation
2. Matrix $A$ is invertible if and only if $0$ is not an eigenvalue (correct answer)
3. Matrix $A$ cannot be invertible since $A^3 = A$ implies $\det(A) = 0$
4. The invertibility depends on whether the eigenvalues are real or complex