Chapter 10.2, Problem 22ES

Draw a graph that has $\left[\begin{array}{ccccc}0& 0& 0& 1& 2\\ 0& 0& 0& 1& 1\\ 0& 0& 0& 2& 1\\ 1& 1& 2& 0& 0\\ 2& 1& 1& 0& 0\end{array}\right]$ as its adjacency matrix. Is this graph biparitite?
Definition: Given an $m\times n$matrix A whose ijth entry is denoted $a_{ij}$ , the transpose of A is the matrix A’ whose ijth entry is $a_{ij}$ , for each $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$.
Note that the first row of A becomes the first column of $A^t$ , the second row of A becomes the second column of $A^t$ , and so forth. For instance.
if $A=\left[\begin{array}{ccc}0& 2& 1\\ 1& 2& 3\end{array}\right]$ , then $A^t=\left[\begin{array}{cc}0& 1\\ 2& 2\\ 1& 3\end{array}\right]$ .

b. Show that a graph with ii vertices is bipartite if, and only if, for some labeling of its vertices, its adjacency matrix has the form $\left[\begin{array}{cc}\text{O}& A\\ A^t& \text{O}\end{array}\right]$ where A is a $k\times \left(n-k\right)$matrix for some integer k such that $0<k<n$, the top left O represents a $k\times k$matrix all of whose entries are 0, $A^t$ is the transpose of A, and the bottom right O represents an $\left(n-k\right)\times \left(n-k\right)$matrix all of whose entries are 0.