Question

Question: Consider a n × nmatrix . Show that swapping the ith andj th rows ofA (where i < j ) amounts to performing 2(j-i)-1 swaps of adjacent rows.

Short answer

Therefore the i-th and j-th row have switched places, and all the in-between rows have returned to their original place, and we had 2(j-i)-1 adjacent row interchanges.

Step-by-step solution

Matrix Definition

Matrix is aset of numbers arranged inrowsandcolumnsso as to form a rectangulararray.

The numbers are called the elements, or entries, of the matrix.

If there are m rows and n columns, the matrix is said to be a “m by n” matrix, written “ m × n.”

To swaps  ith and jth  of A

First, we switch the i-th row with its next row, as many times as it takes to get it to the j-th row.

This takes j-i row interchanges.

Now, we switch the j-th row with its previous row as many times as it takes to get it to the i-th row.

Since it was now in the (j-i)-th row, this took j-i-1 row interchanges.

So, the i-th and j-th row have switched places, and all the in-between rows have returned to their original place, and we had 2(j-i)-1 adjacent row interchanges.