Chapter 16.4, Problem 4OE

To determine

The communication network by using matrix $y$ to the given receive data from the most points.

Answer to Problem 4OE

  $\left[\begin{array}{cccc}1& 0& 1& 1\\ 2& 0& 1& 2\\ 2& 1& 0& 1\\ 2& 1& 1& 0\end{array}\right]$

Explanation of Solution

Given:

The matrix

  $A=\left[\begin{array}{cccc}0& 0& 0& 1\\ 1& 0& 0& 1\\ 1& 1& 0& 0\\ 1& 0& 1& 0\end{array}\right]$

Concept used:

Minors and Cofactors of a square matrix:-

If A is a square matrix then the minor $M_{ij}$ of the entry $a_{ij}$ is the determinant of the matrix obtained by deleting the $i\text{th}$ row and $j\text{th}$ column of $A$

The cofactor $C_{ij}$ of the entry $a_{ij}$ is

  $C_{ij}={\left(-1\right)}^{i+j}{M}_{ij}$

Calculation:

The given matrix

  $A=\left[\begin{array}{cccc}0& 0& 0& 1\\ 1& 0& 0& 1\\ 1& 1& 0& 0\\ 1& 0& 1& 0\end{array}\right]$

The matrix is

  $A+A^2\mathrm{.................}\left(1\right)$

  $\begin{array}{l}{A}^2=\left[\begin{array}{cccc}0& 0& 0& 1\\ 1& 0& 0& 1\\ 1& 1& 0& 0\\ 1& 0& 1& 0\end{array}\right]\left[\begin{array}{cccc}0& 0& 0& 1\\ 1& 0& 0& 1\\ 1& 1& 0& 0\\ 1& 0& 1& 0\end{array}\right]\\ A^2=\left[\begin{array}{cccc}1& 0& 1& 0\\ 1& 0& 1& 1\\ 1& 0& 0& 1\\ 1& 1& 0& 0\end{array}\right]\end{array}$

Putting in equation $\left(1\right)$

  $\begin{array}{l}=\left[\begin{array}{cccc}0& 0& 0& 1\\ 1& 0& 0& 1\\ 1& 1& 0& 0\\ 1& 0& 1& 0\end{array}\right]+\left[\begin{array}{cccc}1& 0& 1& 0\\ 1& 0& 1& 1\\ 1& 0& 0& 1\\ 1& 1& 0& 0\end{array}\right]\\ =\left[\begin{array}{cccc}1& 0& 1& 1\\ 2& 0& 1& 2\\ 2& 1& 0& 1\\ 2& 1& 1& 0\end{array}\right]\end{array}$

Therefore the matrix represents the total receive data from the most points.