Question

Verify that interchanging rows \(p\) and \(q\) \((q>p)\) can be accomplished using \(2(q-p)-1\) adjacent interchanges.

Short answer

The swap requires \(2(q-p)-1\) adjacent interchanges.

Step-by-step solution

Understanding the Problem

We need to verify that interchanging two rows, specifically row \(p\) with row \(q\) (where \(q > p\)), in a matrix can be completed using \(2(q-p)-1\) adjacent interchanges. An adjacent interchange involves swapping consecutive rows.

Counting Adjacent Interchanges for One Swap

Observe that to bring row \(p\) to row \(q\) by adjacent swaps, row \(p\) needs to be moved down one row at a time. Similarly, to move row \(q\) up to row \(p\), it must move one row at a time.

Calculating Total Moves for Row \(p\)

Moving row \(p\) down to the row \(q\) position requires \((q-p)\) adjacent interchanges, as it must pass over each of the rows between \(p\) and \(q\).

Calculating Total Moves for Row \(q\)

Similarly, moving row \(q\) up to the position of row \(p\) requires \((q-p - 1)\) adjacent interchanges, since the row needs to move up past each row between \(q\) and \(p\).

Sum of Moves and Verification

Add the movements calculated: \((q-p) + (q-p-1) = 2(q-p)-1\). Thus, the total number of adjacent interchanges required to swap row \(p\) with row \(q\) is \(2(q-p)-1\), verifying the claim.

Key concepts

Adjacent Interchanges

In matrix operations, especially within linear algebra, understanding adjacent interchanges is crucial. Let's say you need to swap two rows in a matrix, row \(p\) with row \(q\). If \(q > p\), we're interchanging rows that are not immediately next to each other, hence needing multiple swaps.
This concept uses adjacent interchanges, which are swaps involving consecutive rows. Each adjacent interchange affects only two neighboring rows, making larger swaps (like swapping row p and row q) more complex.
A practical way to picture this is to liken each row in a matrix to a train car. Swapping two non-adjacent cars requires moving each car between them individually. As calculated, swapping row \(p\) and row \(q\) requires \(2(q-p)-1\) adjacent interchanges. This total accounts for moving each row exactly the needed number of steps to reorder them correctly.
Understanding why this formula works is essential for mastering row operations in matrices and ensuring efficient computation.

Matrix Manipulation

Matrix manipulation refers to the process of performing operations on matrices to achieve a desired form or result. Among the many operations, row interchanges are often necessary, specifically in solving systems of equations or in finding the inverse of a matrix.
When working with matrices, it is important to remember:

• Matrices are arrays of numbers organized into rows and columns.
• Using operations like row interchanges, you can transform matrices step by step.
• Every row interchange, while simple in concept, can influence the entire matrix. Hence, planning and careful execution are key.

Matrix manipulation is used widely in fields such as computer graphics, statistics, and science. It's all about moving one step closer to solving complex problems. Row interchanges specifically can help simplify the matrix or adjust its form, aiding in analytical evaluation.

Elementary Operations

Elementary operations involve the basic manipulations that can be applied to the rows of matrices. These are the building blocks of more complex matrix transformations. Let's delve into these operations:

• **Swapping two rows:** This is also known as a row interchange, and involves switching the positions of two rows within a matrix. It can be done directly if the rows are adjacent, or through a series of adjacent interchanges if they are not.
• **Multiplying rows by a non-zero scalar:** This operation scales the values of a row, which can be useful for making systems of equations easier to solve.
• **Adding or subtracting a multiple of one row to another:** This operation helps in eliminating variables or simplifying equations in a matrix.

Among these, row swapping is integral for tasks like determining the rank of a matrix. Understanding these operations aids in performing more comprehensive matrix computations smoothly and is essential for students diving into linear algebra.