Problem 12 - Differential Equations and Linear Algebra

Question

Determine the values of the indices \(p\) and \(q\) such that the following are terms in a determinant of order \(4 .\) In each case, determine the number of inversions in the permutation of the column indices and hence find the appropriate sign that should be attached to each term. $$a_{21} a_{3 q} a_{p 2} a_{43}$$.

Step-by-Step Solution

Answer

The values of indices \(p\) and \(q\) are 2 and 4 respectively. There is 1 inversion in the given permutation, which results in a negative sign for the term in the determinant of order \(4\). Hence, the term with the appropriate sign will be: \(-a_{21} a_{34} a_{22} a_{43}\).

Step 1: All 4! permutations of column indices

We need to list all possible permutations of column indices (1, 2, 3, 4). There are 4! = 4 × 3 × 2 × 1 = 24 permutations. Step 2: Find the permutation of column indices for which the given term is present

Step 2: Identifying the permutation of column indices for the given term

Given term: \(a_{21} a_{3q} a_{p2} a_{43}\). We need to find the permutation in which this term is present. From the given term, we can extract the following column indices: - \(a_{21}\): column 1 - \(a_{3q}\): column \(q\) - \(a_{p2}\): column 2 - \(a_{43}\): column 3 Sequence of column indices: (1, q, 2, 3). Since \(q\) cannot be 1, 2, or 3 (because all are already present in the sequence), it must be 4. Therefore, the actual sequence is (1, 4, 2, 3). Now we need to find the index for row \(p\), so the given term should be written as \(a_{21} a_{34} a_{p2} a_{43}\). Row \(p\) must be 2, therefore \(p = 2\). The permutation of the column indices should be (1, 4, 2, 3). Step 3: Determine the number of inversions and the appropriate sign

Step 3: Number of Inversions

To find the number of inversions in the permutation of column indices (1, 4, 2, 3), we need to count the number of times we have an element followed by a smaller element in the sequence. - First inversion: 1 followed by 4: No inversions - Second inversion: 4 followed by 2: 1 inversion - Third inversion: 2 followed by 3: No inversions - Fourth inversion: 1 followed by 2: No inversions Total number of inversions = 1

Step 4: Appropriate Sign

Since there is an odd number of inversions (1 inversion), the appropriate sign for the given term will be negative (-). The values of indices \(p\) and \(q\) are found to be 2 and 4 respectively. There is 1 inversion in the given permutation, which results in a negative sign for the term in the determinant of order \(4\). Hence, the term with the appropriate sign will be: \(-a_{21} a_{34} a_{22} a_{43}\).

Key Concepts

Permutation of Column Indices

In linear algebra, when determining the value of a determinant, the concept of the permutation of column indices is vital. A permutation is a specific arrangement of objects in a particular order. In the context of the determinant of a matrix, we refer to the arrangement of the column indices.

For a determinant of order 4, like in our exercise, there are 4 factorial (4!) possible permutations of the column indices, resulting in 24 distinct permutations. When evaluating the terms in a determinant, it's crucial not only to consider the values of the elements but also their positions, which are determined by the permutation of the indices.

An easy way to understand this is by picturing the determinant as a grid, where the position of each element is given by its row and column numbers. Only one element from each column (and each row) can be part of a term in the determinant. This leads us to identify the unique permutation for the columns in any given term.

For example, the term in our exercise has four factors drawn from specific rows and columns. We must then determine the specific arrangement of column indices that allows us to select those factors. This process of determining which permutation of column indices corresponds to a specific term is essential in evaluating determinants, as it lays the groundwork for further steps such as calculating the number of inversions and the sign of the term.

Number of Inversions

The number of inversions in a permutation is a concept that often puzzles students, yet it's a critical aspect of calculating the sign of a determinant's terms. An inversion is a pair of elements in a sequence where the larger number precedes a smaller number.

To count the number of inversions, we compare each element with the elements following it and count instances where the later element is smaller. In simple terms, you're looking for times the sequence goes 'downhill.' The significance of an inversion is rooted in its contribution to the determinant's sign.

Let's revisit our exercise, focusing exclusively on the sequence of column indices (1, 4, 2, 3). When we inspect this permutation, we look for instances where an index precedes indices that are smaller than itself. We find that 4 precedes 2, so that's one inversion right there. The other pairs do not create inversions because they either maintain the increasing order or have elements that do not directly follow each other.

The total count of inversions will then guide us towards the sign of the term - an odd count signifies a negative sign, while an even count leads to a positive sign. Understanding how to count inversions accurately is a foundational skill for anyone learning about determinants in linear algebra.

Sign of Terms in a Determinant

The sign of terms in a determinant adds another layer of complexity to understanding determinants in linear algebra. It is directly influenced by the number of inversions we just discussed. Once we have figured out the number of inversions in a specific permutation of the column indices, determining the sign is straightforward.

If the number of inversions is odd, the sign attached to the term will be negative; conversely, if the number of inversions is even, the sign is positive. Remembering this rule can be simplified as an inversion parity check – like checking whether a number is odd or even.

In our example, we determined that there was a single inversion in the permutation of column indices (1, 4, 2, 3). Following our rule, this means that the sign of the term is negative. To correctly represent the term within the determinant, we place a negative sign before the product of elements selected from those rows and columns, hence yielding -(a_{21} a_{34} a_{22} a_{43}).

Getting comfortable with determining the sign of terms will help you skillfully approach a wide range of problems in linear algebra. It's an essential step in ensuring the accuracy of the determinant's value and can't be overlooked when solving more complex algebraic expressions.