Question

Consider an arbitrary currency exchange matrixA. See Exercises 60and 61.

a. What are the diagonal entries ${\mathit{a}}_{\mathbf{i}\mathbf{i}}$of A?

b. What is the relationship between ${\mathit{a}}_{\mathbf{i}\mathbf{j}}$and ${\mathit{a}}_{\mathbf{j}\mathbf{i}}$?

c. What is the relationship among ${\mathit{a}}_{\mathbf{i}\mathbf{k}}\mathbf{,}{\mathit{a}}_{\mathbf{k}\mathbf{j}}$, and ${\mathit{a}}_{\mathbf{i}\mathbf{j}}$?

d. What is the rank of A? What is the relationship betweenAand $\mathit{r}\mathit{r}\mathit{e}\mathit{f}\mathbf{\left(}\mathit{A}\mathbf{\right)}$?

Short answer

a. The diagonal elements of $A=\left[\begin{array}{cc}1& 1/8\\ 8& 1\end{array}\right]$are all $1\text{'}s$. The diagonal elements of $A=\left[\begin{array}{cccc}1& \frac{8}{10}& \frac{1}{8}& \frac{10}{8}\\ \frac{10}{8}& 1& \frac{5}{32}& \frac{25}{16}\\ 8& \frac{32}{5}& 1& 10\\ \frac{8}{10}& \frac{16}{25}& \frac{1}{10}& 1\end{array}\right]$are all$1\text{'}s$.

b. The values of $a_{ij}$and $a_{ji}$are reciprocal of each other.

c. The relation between $a_{ij}$, $a_{ik}$and $a_{kj}$is, $a_{ij}=a_{ik}\times a_{kj}$, that is, one element is product of other two.

d. $\text{Rank\hspace{0.17em}}\left(\begin{array}{cc}1& \frac{1}{8}\\ 8& 1\end{array}\right)=1$and$\text{Rank}\left(\begin{array}{cccc}1& \frac{8}{10}& \frac{1}{8}& \frac{10}{8}\\ \frac{10}{8}& 1& \frac{5}{32}& \frac{25}{16}\\ 8& \frac{32}{5}& 1& 10\\ \frac{8}{10}& \frac{16}{25}& \frac{1}{10}& 1\end{array}\right)=1$

The rank and the rref represents the value one.

Step-by-step solution

Compute the diagonal elements.

(a)

The matrices and their diagonals are,

$A=\left[\begin{array}{cc}1& 1/8\\ 8& 1\end{array}\right]$

The diagonal elements are, 1

$A=\left[\begin{array}{cccc}1& \frac{8}{10}& \frac{1}{8}& \frac{10}{8}\\ \frac{10}{8}& 1& \frac{5}{32}& \frac{25}{16}\\ 8& \frac{32}{5}& 1& 10\\ \frac{8}{10}& \frac{16}{25}& \frac{1}{10}& 1\end{array}\right]$

The diagonal elements are, 1.

Compute the values of the components.

(b)

The values of the elements are,

$A=\left[\begin{array}{cc}1& 1/8\\ 8& 1\end{array}\right]$

The values are, $a_{12}=\frac{1}{8},a_{21}=8$

$A=\left[\begin{array}{cccc}1& \frac{8}{10}& \frac{1}{8}& \frac{10}{8}\\ \frac{10}{8}& 1& \frac{5}{32}& \frac{25}{16}\\ 8& \frac{32}{5}& 1& 10\\ \frac{8}{10}& \frac{16}{25}& \frac{1}{10}& 1\end{array}\right]$

The values are,

$a_{12}=\frac{8}{10}\Rightarrow a_{21}=\frac{10}{8}$

$a_{34}=10\Rightarrow a_{43}=\frac{1}{10}$

$a_{41}=\frac{8}{10}\Rightarrow a_{14}=\frac{10}{8}$

$a_{24}=\frac{25}{16}\Rightarrow a_{42}=\frac{16}{25}$

$a_{31}=8\Rightarrow a_{13}=\frac{1}{8}$

$a_{32}=\frac{32}{5}\Rightarrow a_{23}=\frac{5}{32}$

Therefore, the values of $a_{ij}$and $a_{ji}$are reciprocal of each other.

Compute the relation between the elements.

(c)

Consider the matrix,

$A=\left[\begin{array}{cccc}1& \frac{8}{10}& \frac{1}{8}& \frac{10}{8}\\ \frac{10}{8}& 1& \frac{5}{32}& \frac{25}{16}\\ 8& \frac{32}{5}& 1& 10\\ \frac{8}{10}& \frac{16}{25}& \frac{1}{10}& 1\end{array}\right]$

The values of the elements are,

$a_{11}=a_{12}\times a_{21}\Rightarrow a_{11}=\frac{8}{10}\times \frac{10}{8}=1$

$a_{22}=a_{21}\times a_{12}\Rightarrow a_{22}=\frac{10}{8}\times \frac{8}{10}=1$

$a_{44}=a_{43}\times a_{34}\Rightarrow a_{44}=\frac{1}{10}\times 10=1$

$a_{24}=a_{21}\times a_{14}\Rightarrow a_{24}=\frac{10}{8}\times \frac{10}{8}=\frac{25}{16}$

$a_{31}=a_{34}\times a_{41}\Rightarrow a_{31}=10\times \frac{8}{10}=8$

$a_{32}=a_{31}\times a_{12}\Rightarrow a_{32}=8\times \frac{8}{10}=\frac{32}{5}$

$a_{33}=a_{31}\times a_{13}\Rightarrow a_{33}=8\times \frac{1}{8}=1$

The relation between $a_{ij}$, $a_{ik}$and $a_{kj}$is, $a_{ij}=a_{ik}\times a_{kj}$, that is, one element is product of other two.

Compute the rank and rref of matrices.

(d)

Now, we have

$\text{Rank\hspace{0.17em}}\left(\begin{array}{cc}1& \frac{1}{8}\\ 8& 1\end{array}\right)=1,rref\left(\begin{array}{cc}1& \frac{1}{8}\\ 8& 1\end{array}\right)=\left(\begin{array}{cc}8& 1\\ 0& 0\end{array}\right)$and$\text{Rank\hspace{0.17em}}\left(\begin{array}{cccc}1& \frac{8}{10}& \frac{1}{8}& \frac{10}{8}\\ \frac{10}{8}& 1& \frac{5}{32}& \frac{25}{16}\\ 8& \frac{32}{5}& 1& 10\\ \frac{8}{10}& \frac{16}{25}& \frac{1}{10}& 1\end{array}\right)=1,rref\left(\begin{array}{cccc}1& \frac{8}{10}& \frac{1}{8}& \frac{10}{8}\\ \frac{10}{8}& 1& \frac{5}{32}& \frac{25}{16}\\ 8& \frac{32}{5}& 1& 10\\ \frac{8}{10}& \frac{16}{25}& \frac{1}{10}& 1\end{array}\right)=\left(\begin{array}{cccc}1& \frac{4}{5}& \frac{1}{8}& \frac{5}{4}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$

The rank of the matrices is one. The reduced row-echelon form of the matrices have single row.