Quiz 3 Review Questions

Short practice problems by topic

Reviews

Cramer’s Rule

Questions

  1. Solve using Cramer’s rule: \[ \begin{cases} x+y=4\\ 2x-y=1 \end{cases} \]

  2. For what value(s) of \(k\) does Cramer’s rule apply to \[ \begin{cases} x+2y=3\\ kx+4y=1 \end{cases} \]

  3. Suppose for a \(3\times 3\) system \(Ax=b\), \[ \det(A)=5,\qquad \det(A_1)=10,\qquad \det(A_2)=-15,\qquad \det(A_3)=20. \] Find the solution.

  4. True or false: If \(\det(A)=0\), then the system \(Ax=b\) has no solution.

Solutions

  1. \[ x=\frac{5}{3},\qquad y=\frac{7}{3}. \]

  2. Cramer’s rule applies when \[ \det\begin{pmatrix}1&2\\k&4\end{pmatrix}=4-2k\neq 0, \] so \[ k\neq 2. \]

  3. \[ x_1=\frac{10}{5}=2,\qquad x_2=\frac{-15}{5}=-3,\qquad x_3=\frac{20}{5}=4. \]

  4. False. If \(\det(A)=0\), the system may have no solution or infinitely many solutions.

Cross Product / Determinant Applications

Questions

  1. Compute \[ (1,0,2)\times(0,1,1). \]

  2. Find the area of the parallelogram spanned by \[ u=(1,0,0),\qquad v=(0,2,0). \]

  3. (Optional) Find a normal vector to the plane containing \[ P=(0,0,0),\quad Q=(1,1,0),\quad R=(1,0,1). \]

  4. True or false: If \(u\times v=0\), then \(u\) and \(v\) are perpendicular.

Solutions

  1. \[ (1,0,2)\times(0,1,1)=(-2,-1,1). \]

  2. \[ u\times v=(0,0,2),\qquad \|u\times v\|=2. \] So the area is \(2\).

  3. Use \[ Q-P=(1,1,0),\qquad R-P=(1,0,1). \] Then \[ (Q-P)\times(R-P)=(1,-1,-1). \] Any nonzero scalar multiple is also a valid normal vector.

  4. False. It means the vectors are parallel (or one is the zero vector).

Eigenvalues and Eigenvectors

Questions

  1. Find the eigenvalues of \[ A=\begin{pmatrix} 4&1\\ 0&2 \end{pmatrix}. \]

  2. Find an eigenvector of \[ A=\begin{pmatrix} 3&0\\ 0&-1 \end{pmatrix} \] for eigenvalue \(\lambda=-1\).

  3. Let \[ A=\begin{pmatrix} 2&0\\ 0&5 \end{pmatrix}. \] Describe geometrically what \(A\) does to vectors in the \(x\)- and \(y\)-directions.

  4. True or false: Every nonzero vector is an eigenvector of the identity matrix \(I\).

  5. Further practice questions are on WebAssign.

Solutions

  1. Since \(A\) is triangular, the eigenvalues are the diagonal entries: \[ 4,\ 2. \]

  2. One eigenvector is \[ \begin{pmatrix}0\\1\end{pmatrix}. \]

  3. The matrix scales vectors in the \(x\)-direction by \(2\) and vectors in the \(y\)-direction by \(5\). These coordinate directions are eigenvector directions.

  4. True. For every nonzero vector \(v\), \[ Iv=v=1\cdot v, \] so every nonzero vector is an eigenvector with eigenvalue \(1\).

Diagonalization

Questions

  1. A matrix \(A\) has eigenvectors \[ v_1=\binom{1}{0},\qquad v_2=\binom{1}{1} \] with eigenvalues \(2\) and \(5\), respectively. Write down \(P\) and \(D\) so that \[ A=PDP^{-1}. \]

  2. True or false: A \(3\times 3\) matrix with three distinct eigenvalues is diagonalizable.

  3. Is the matrix \[ \begin{pmatrix} 2&1\\ 1&2 \end{pmatrix} \] guaranteed to be diagonalizable over \(\mathbb R\)? Why?

  4. True or false: Every diagonalizable matrix is symmetric.

  5. Further practice questions are on WebAssign.

Solutions

  1. \[ P=\begin{pmatrix}1&1\\0&1\end{pmatrix}, \qquad D=\begin{pmatrix}2&0\\0&5\end{pmatrix}. \]

  2. True.

  3. Yes. It is real symmetric, so by the spectral theorem it is orthogonally diagonalizable.

  4. False. Symmetric matrices are diagonalizable, but not every diagonalizable matrix is symmetric.

Solving ODEs Using Matrix Exponential

Questions

  1. Solve \[ x'(t)=Ax(t),\qquad A=\begin{pmatrix}2&0\\0&-1\end{pmatrix}, \qquad x(0)=\binom{3}{1}. \]

  2. Let \[ A=PDP^{-1},\qquad D=\begin{pmatrix}1&0\\0&4\end{pmatrix}. \] What is \(e^{tD}\)?

  3. If \(v\) is an eigenvector of \(A\) with eigenvalue \(\lambda\), solve \[ x'(t)=Ax(t),\qquad x(0)=v. \]

  4. (Optional) Let \[ N=\begin{pmatrix}0&1\\0&0\end{pmatrix}. \] Compute \(e^{tN}\).

Solutions

  1. \[ x(t)=\binom{3e^{2t}}{e^{-t}}. \]

  2. \[ e^{tD}=\begin{pmatrix}e^t&0\\0&e^{4t}\end{pmatrix}. \]

  3. \[ x(t)=e^{\lambda t}v. \]

  4. Since \(N^2=0\), \[ e^{tN}=I+tN=\begin{pmatrix}1&t\\0&1\end{pmatrix}. \]

Singular Value Decomposition (SVD)

Questions

  1. True or false: Every real matrix has an SVD.

  2. True or false: Singular values can be negative.

  3. What are the singular values of \[ A=\begin{pmatrix}3&0\\0&1\end{pmatrix}? \]

  4. Is SVD the same as diagonalization for every diagonalizable matrix?

Solutions

  1. True.

  2. False. Singular values are always nonnegative.

  3. The singular values are \[ 3,\ 1. \]

  4. No. Diagonalization uses eigenvectors and may fail to exist, while SVD always exists and uses orthonormal singular vectors.

Concept Reminders

Short conceptual answers

  • When does Cramer’s rule apply? It applies exactly when the coefficient matrix is invertible, i.e. when \(\det(A)\neq 0\).

  • What does the cross product tell you geometrically? The vector \(u\times v\) is perpendicular to both \(u\) and \(v\), and \(\|u\times v\|\) is the area of the parallelogram they span.

  • What is the geometric meaning of an eigenvector? It is a direction preserved by the linear transformation; the vector is only stretched, compressed, or flipped.

  • What are two easy sufficient conditions for diagonalizability? A matrix is diagonalizable if it has \(n\) distinct eigenvalues, and every real symmetric matrix is diagonalizable by the spectral theorem.

  • Why is diagonalization useful for ODEs? If \(A=PDP^{-1}\), then \[ e^{tA}=Pe^{tD}P^{-1}, \] and exponentiating a diagonal matrix is easy.

  • How is SVD different from diagonalization? Diagonalization is \(A=PDP^{-1}\) and requires enough eigenvectors, while SVD is \(A=U\Sigma V^T\) and exists for every matrix.