Problem 10 - Differential Equations and Linear Algebra

Question

Determine whether the given expression is a term in the determinant of order \(5 .\) If it is, determine whether the permutation of the column indices has even or odd parity and hence find whether the term has a plus or a minus sign attached to it. $$a_{11} a_{32} a_{24} a_{43} a_{55}$$.

Step-by-Step Solution

Answer

The given expression \(a_{11} a_{32} a_{24} a_{43} a_{55}\) is a term in the determinant of order 5. The permutation of the column indices has even parity which results in a positive sign for the term. Therefore, the term is \(+a_{11} a_{32} a_{24} a_{43} a_{55}\).

Step 1: Determining Uniqueness of Rows and Columns

Let's check if the expression \(a_{11} a_{32} a_{24} a_{43} a_{55}\) contains unique elements from each row and column. We can do this by looking at the row and column indices for each element: - \(a_{11}\): Row 1, Column 1 - \(a_{32}\): Row 3, Column 2 - \(a_{24}\): Row 2, Column 4 - \(a_{43}\): Row 4, Column 3 - \(a_{55}\): Row 5, Column 5 Since all rows (1, 2, 3, 4, 5) and all columns (1, 2, 3, 4, 5) appear exactly one time, we can conclude that the expression is a term in the determinant of order 5.

Step 2: Determining Parity of Permutation

Now, let's find out whether the permutation of the column indices has even or odd parity. The column indices in the given expression are (1, 2, 4, 3, 5). We can find the inversions in the permutation as follows: 1. (1, 2) has 0 inversions since 1 < 2. 2. (1, 4) has 0 inversions since 1 < 4. 3. (1, 3) has 0 inversions since 1 < 3. 4. (1, 5) has 0 inversions since 1 < 5. 5. (2, 4) has 0 inversions since 2 < 4. 6. (2, 3) has 1 inversion since 2 > 3. 7. (2, 5) has 0 inversions since 2 < 5. 8. (4, 3) has 1 inversion since 4 > 3. 9. (4, 5) has 0 inversions since 4 < 5. 10. (3, 5) has 0 inversions since 3 < 5. We have a total of 2 inversions, which is an even number. Therefore, this permutation has even parity.

Step 3: Determining the Sign of the Term

Since the permutation of the column indices has even parity, this term in the determinant of order 5 will have a positive (+) sign attached to it. The term with a plus sign is \(+a_{11} a_{32} a_{24} a_{43} a_{55}\).

Key Concepts

Determinant of a Matrix

Understanding the determinant of a matrix is vital in linear algebra. It is a scalar value that can be computed from the elements of a square matrix and provides significant information about the matrix. For instance, it can tell us if the matrix is invertible (a non-zero determinant) or if the system of equations corresponding to the matrix has a unique solution.

The determinant of a matrix is calculated by adding and subtracting products of its elements according to a specific pattern. For a 5x5 matrix, as seen in the exercise, we consider all the possible products that include exactly one element from each row and each column. The expression given in the problem, \(a_{11} a_{32} a_{24} a_{43} a_{55}\), is one such term of the matrix's determinant. The terms are then combined with a plus or minus sign depending on the parity of the permutation of the column indices corresponding to each term.

Permutation Parity

Permutation parity refers to whether the sequence of numbers is ordered in an even or odd manner. In other words, it reflects the number of swaps needed to arrange a sequence into its natural order (from smallest to largest). If the number of swaps is even, the permutation is said to have even parity; if the number is odd, it is said to have odd parity.

Permutation parity is essential when determining the sign to attach to a term when calculating the determinant of a matrix. The term from our example, \(a_{11} a_{32} a_{24} a_{43} a_{55}\), necessitates examining the order of column indices and counting inversions—which are pairs of numbers out of their natural sequence—to determine its parity.

Inversions in Permutations

When we talk about permutations in linear algebra, inversions are instances where a pair of elements are out of their natural order. To be specific, in a sequence, if we have two elements where the first is greater than the second, and it comes before the second when the sequence is laid out in increasing order, this is called an inversion.

In the exercise given, we observe that the column sequence (1, 2, 4, 3, 5) has two inversions: (2, 3) and (4, 3). The quantity of these inversions is crucial because it determines the permutation's parity. A simple yet effective way of checking inversions is to compare each pair in the sequence, as demonstrated in the step-by-step solution.

Sign of Determinant Terms

The sign of terms in the determinant expression is determined by the permutation's parity. This rule is a part of the Leibniz formula for determinants, which states that each term's sign is positive if the permutation of its indices has even parity and negative if the parity is odd.

Therefore, the term we are analyzing in the exercise, as a part of the determinant, will inherit its sign from the parity of the permutation. Since the permutation of the column indices (1, 2, 4, 3, 5) in the term \(a_{11} a_{32} a_{24} a_{43} a_{55}\) has even parity, based on the two inversions identified, the term will have a positive (+) sign.