General linear algebra questions

The following problems are “matrix algebra” review for students in upper level undergrad courses, such as math 519 (differential equations). There it is assumed that students have learned this material in courses such as math 375:
  1. Suppose the $3\times3$ matrix $A$ has eigenvectors
    $u_1=\begin{pmatrix} 1\\0\\0 \end{pmatrix}$,   $u_2=\begin{pmatrix} 1\\1\\0 \end{pmatrix}$,   $u_3=\begin{pmatrix} 1\\0\\1 \end{pmatrix}$
    with corresponding eigenvalues $\lambda_1=2$, $\lambda_2=\frac12$, and $\lambda_3=-\frac12$.
    1. Compute the determinant and trace of $A$. Explain your answers.
    2. Is $A$ invertible? Explain.
    3. If $y=\begin{pmatrix}4\\1\\2\end{pmatrix}$, then solve $Ax=y$.
    4. Find $A^{2017} z$ where $z=\begin{pmatrix}0\\0\\1\end{pmatrix}$.
  2. Let $A, B$ be $n\times n$ matrices, and let $I$ be the $n\times n$ identity matrix. Are the following True or False? Remember: true means “always true,” i.e. for any choice of matrices $A,B$. For those statements that are false provide an example showing why they are not always true; for the true statements provide a reason.
    1. $A+B=B+A$ ?
    2. $AB = BA$ ?
    3. $(A+B)^2 = A^2 + 2AB + B^2$ ?
    4. $(I+A)^2 = I + 2A + A^2$ ?
    5. If $A\vv=B\vv$ for some vector $\vv\in\R^n$ then $A=B$ ?
    6. If $A\vv=B\vv$ for all vectors $\vv\in\R^n$ then $A=B$ ?
  3. Suppose that $A$ and $B$ are $n\times n$ matrices that commute, i.e. for which $AB=BA$.
    1. Suppose that $A$ is invertible. Show that $A^{-1}B=BA^{-1}$.
    2. If $\vv$ is an eigenvector with eigenvalue $\lambda$ for the matrix $A$, and if $B\vv\neq0$, then show that $B\vv$ also is an eigenvector, also with eigenvalue $\lambda$ for $A$.
  4. Let $A$ be a $5\times 5$ matrix, and let $\vv$, $\vw$ be eigenvectors of $A$ with eigenvalues $\lambda=2$, and $\mu=-3.$ Let $I$ be the $5\times 5$ identity matrix.
    1. Can $\vv=0$? What about $\vw$? Explain.
    2. Compute $A(2\vv-2\vw)$, $A^2(2\vv-2\vw)$, and $A^3(2\vv-2\vw)$
    3. Compute $(A+I)(2\vv-2\vw)$.
    4. Compute $(A+I)^2(2\vv-2\vw)$.
    5. Compute $(A-2I)(A+3I)(2\vv-2\vw)$.
    6. Suppose $s\vv+t\vw=0$ for certain numbers $s, t\in\R$. Show directly that $s=t=0$ by expanding $A(s\vv+t\vw)$.
    7. Are $\vv, \vw $ linearly independent? Explain.
    8. Is $\{\vv, \vw\}$ a basis for $\R^5$? Explain.
  5. Let $A$ be a $3\times3$ matrix with eigenvalues $\lambda=1$, $\mu=-1$, $\nu=2$, and corresponding eigenvectors $\vu, \vv, \vw$.
    1. Are $\vu, \vv, \vw$ linearly independent? Explain.
    2. Is $\{\vu, \vv, \vw\}$ a basis for $\R^3$? Explain.
    3. Explain why any vector $\vx$ can be written as a linear combination of $\vu, \vv, \vw$.
    4. Solve the equation $A\vx = 2\vu+3\vv-\vw$ for $\vx$ (suggestion: look for a solution $\vx$ that is a linear combination of $\vu$, $\vv$, and $\vw$.)