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Chapter 6: Q67E (page 293)

Consider a patternP in an脳n matrix, and choose an entryaijin this pattern. Show that the number of inversions involvingaij is even if (i+j) is even and odd if (i+j) is odd. Hint: Suppose there arek entries in the pattern to the left and above aij . Express the number of inversions involvingaij in terms ofk.

Short Answer

Expert verified

Therefore, the number of inversions including aijis i-1-k+j-1-k=i+j-2-2k, which is even if i+j is even, and odd if i+j is odd.

Step by step solution

01

Matrix Definition 

A matrix is a set of numbers arranged in rows and columns so 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 鈥�m by n 鈥� matrix, written 鈥� m脳n.鈥�

02

To find the number of inversions involving  aij in terms of k

Consider k to be the number of entries, in this pattern, that are to the left and above aij.

Clearly then, the number of the pattern's entries to the right and above aijis i-1-k.

In a similar manner, the number of such entries to the left and below aij is j-1-k.

Therefore, the number of inversions including aijisrole="math" localid="1660723822760" i-1-k+j-1-k=i+j-2-2k, which is even if i+j is even, and odd if i+j is odd.

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