Practice Test 3 - Linear Algebra

20 questions. Answer them in any order, then submit to see your score and every explanation.

Start practice test

1. A population is modeled by the system $\mathbf{x}_{k+1} = A\mathbf{x}_k$, where $A$ is a diagonalizable matrix with eigenvalues $\lambda_1 = 1.1$ and $\lambda_2 = 0.8$, and corresponding eigenvectors $\mathbf{v}_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$ and $\mathbf{v}_2 = \begin{pmatrix} -1 \\ 2 \end{pmatrix}$. If the initial population is $\mathbf{x}_0 = \begin{pmatrix} 5 \\ 4 \end{pmatrix}$, what is the approximate state of the system for very large $k$?
2. For what values of the constant $k$ is the matrix $A = \begin{pmatrix} 1 & k & 0 \\ 2 & 4 & 1 \\ 3 & 5 & k \end{pmatrix}$ non-invertible?
3. A $3 \times 3$ real matrix $A$ must have at least one real eigenvalue. Which of the following sets could NOT be the complete set of eigenvalues for such a matrix $A$?
4. A $2 \times 2$ matrix $A$ is transformed by the row operation $R_1 - 2R_2 \to R_1$ to produce the matrix $B = \begin{pmatrix} 1 & 0 \\ 3 & 1 \end{pmatrix}$. What is the original matrix $A$?
5. Let $\vec{v}_1 = \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}$, $\vec{v}_2 = \begin{pmatrix} -2 \\ 3 \\ -1 \end{pmatrix}$, and $\vec{v}_3 = \begin{pmatrix} 1 \\ 0 \\ k \end{pmatrix}$. For which value of $k$ does the set $\{\vec{v}_1, \vec{v}_2, \vec{v}_3\}$ NOT form a basis for $\mathbb{R}^3$?
6. For what value of $k$ does the following system of equations have no solution? \begin{align*} x + y - z &= 1 \\ 2x + 3y + kz &= 3 \\ x + ky + 3z &= 2 \end{align*}
7. Let $S = \{\vec{v_1}, \vec{v_2}, \vec{v_3}\}$ and $T = \{\vec{u_1}, \vec{u_2}, \vec{u_3}\}$ be two bases for a vector space $V$. If the change of basis matrix $P$ from $S$ to $T$ satisfies $P\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} = \begin{pmatrix} 3 \\ 0 \\ 2 \end{pmatrix}$, what can we conclude?
8. A transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ can be represented by the matrix equation $T(\mathbf{x}) = A\mathbf{x}$ for some matrix $A$. Which statement about the relationship between this matrix representation and linearity is most accurate?
9. Matrix $A$ has the property that $A^4 = 2A^2$. Which statement about the possible values of $\det(A)$ is correct?
10. Consider the set $S = \{(x,y) \in \mathbb{R}^2 \mid x^2 - y^2 = 0\}$. Which of the following claims about $S$ is true?
11. Let $A$ be a general $3 \times 3$ matrix with elements $a_{ij}$ and cofactors $C_{ij}$. Which expression correctly represents the cofactor expansion of $\det(A)$ along the second column?
12. What is the determinant of the matrix $B = \begin{pmatrix} 2 & -1 & 3 \\ 4 & -2 & 6 \\ 1 & 5 & -7 \end{pmatrix}$?
13. Let $M$ be a $4 \times 4$ matrix with characteristic polynomial $p(\lambda) = (\lambda - 1)^2(\lambda - 3)^2$. Suppose that $\dim(\text{null}(M - I)) = 1$ and $\dim(\text{null}((M - I)^2)) = 2$. What can be concluded about the Jordan canonical form of $M$?
14. Let $T: \mathbb{R}^n \to \mathbb{R}^n$ be a linear transformation. If $T$ is one-to-one, which of the following must also be true?
15. Let $A$ be a $3 \times 5$ matrix with $\mathrm{rank}(A) = 2$. What is the nullity of the transpose matrix, $A^T$?
16. A Markov process is described by the transition matrix $P = \begin{pmatrix} 0.5 & 0.25 \\ 0.5 & 0.75 \end{pmatrix}$. Which of the following vectors is the steady-state vector $\vec{q}$ for this process?
17. Let $\vec{u}$ and $\vec{v}$ be two non-zero, non-parallel vectors in $\mathbb{R}^3$. Let $\vec{w} = 2\vec{u} - 3\vec{v}$. Which geometric object best describes the set of all linear combinations of the vectors in $S = \{\vec{u}, \vec{v}, \vec{w}\}$?
18. A researcher sets up normal equations to solve an overdetermined linear system, but discovers that the coefficient matrix $A^TA$ is singular. Given that the original matrix $A$ is $4 \times 3$, what is the most likely explanation for this singularity, and what does it imply about the solution?
19. If $A$ is a $3 \times 4$ matrix with singular values $\sigma_1 = 7$, $\sigma_2 = 5$, $\sigma_3 = 0$, what is the dimension of the null space of $A^T$?
20. Let $A$ be a $5 \times 3$ matrix. A basis for the null space of $A^T$ is given by $\left\{ \begin{pmatrix} 1 \\ 0 \\ 2 \\ 0 \\ -1 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ -1 \\ 3 \\ 0 \end{pmatrix} \right\}$. For which vector $b$ is the system $Ax = b$ guaranteed to be consistent?

Practice tests