Question 1205503: Express the following as a product of transposition in two ways.
1. [1 2 2 1 3 4 4 3 5 6 6 5 7 8 8 7 ]
2. [1 2 2 1 3 4 4 5 5 6 3 7 7 8 8 6 ]
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! To express the given matrices as a product of transpositions, we need to identify the pairs of elements that need to be swapped to transform the identity matrix into the given matrix.
**1. [1 2 2 1 3 4 4 3 5 6 6 5 7 8 8 7 ]**
This matrix can be obtained by swapping the following pairs: (1,2), (3,4), (5,6), and (7,8).
Therefore, we can express the matrix as a product of transpositions in two ways:
**Way 1:**
```
(1 2)(3 4)(5 6)(7 8)
```
**Way 2:**
```
(7 8)(5 6)(3 4)(1 2)
```
**2. [1 2 2 1 3 4 4 5 5 6 3 7 7 8 8 6 ]**
This matrix can be obtained by swapping the following pairs: (1,2), (3,4), (5,6), and (3,7).
Therefore, we can express the matrix as a product of transpositions in two ways:
**Way 1:**
```
(3 7)(1 2)(3 4)(5 6)
```
**Way 2:**
```
(5 6)(3 4)(1 2)(3 7)
```
Note that there are many other ways to express these matrices as products of transpositions. The above are just two examples.
|
|
|
