# Math 487, Homework 1

$latex \def\Real{\mathbb{R}}\def\ee{\mathbf{e}}\def\ff{\mathbf{f}}\newcommand\ket[1]{\vert{#1}\rangle}\newcommand\bra[1]{\langle{#1}\vert}\newcommand\bk[2]{\langle #1\vert#2\rangle}\def\lin{\mathtt{Lin}}$

1. (15) Find all matrices
$$X=\left( \begin{array}{cc} x&y\\ z&w \end{array} \right)$$
such that $latex XA=BX$, where
$$A=\left( \begin{array}{cc} 1&2\\ -1&0 \end{array} \right), \mathrm{\ and\ } B=\left( \begin{array}{cc} 0&1\\ 3&0 \end{array} \right).$$
2. (15) Same for
$$A=\left( \begin{array}{cc} 2&0\\ 0&0 \end{array} \right), \mathrm{\ and\ } B=\left( \begin{array}{cc} 1&3\\ 1&-1 \end{array} \right).$$
3. (20) Find $latex LU$ decomposition for the Vandermonde matrices of sizes 2,3,4:
$$V_2=\left( \begin{array}{cc} 1&1\\ x_1&x_2 \end{array} \right); V_3=\left( \begin{array}{ccc} 1&1&1\\ x_1&x_2&x_3\\ x^2_1&x^2_2&x^2_3 \end{array} \right), \mathit{etc.}$$
4. (20) Find matrix $latex A$ (of smallest possible size), such thatÂ  $latex A^3\neq 0, A^4=0$.
5. (30) Find the general formula for the determinant of the tridiagonal $latex n\times n$ matrix
$$\det\left( \begin{array}{ccccc} 3& 2& 0&\ldots\\ 1& 3& 2&\ldots\\ 0& 1& 3&\ldots\\ \vdots&\vdots&\vdots&\ddots \end{array} \right)$$