Problem 14 - Differential Equations and Linear Algebra

Question

Evaluate the determinant of the given matrix by first using elementary row operations to reduce it to upper triangular form. $$\begin{array}{|lllll|} 1 & 2 & 3 & 4 & 5 \\ 5 & 4 & 3 & 2 & 1 \\ 2 & 3 & 4 & 5 & 6 \\ 6 & 5 & 4 & 3 & 2 \\ 0 & 2 & 4 & 6 & 8 \end{array} |$$

Step-by-Step Solution

Answer

The determinant of the given matrix A is 1, after performing the Gaussian elimination process using row operations to convert the matrix A into an upper triangular matrix, and then calculating the product of the diagonal elements.

Step 1: Set up the 5x5 matrix

First, let's set up the given matrix A: \[ A = \begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 5 & 4 & 3 & 2 & 1 \\ 2 & 3 & 4 & 5 & 6 \\ 6 & 5 & 4 & 3 & 2 \\ 0 & 2 & 4 & 6 & 8 \end{bmatrix} \]

Step 2: Apply row operations

Now, we will perform the Gaussian elimination process by using row operations to convert the matrix A into an upper triangular matrix. We can use these row operations: 1. Swap two rows (denoted as R1 <-> R2) 2. Multiply a row by a nonzero scalar (denoted as kR1) 3. Add or subtract a multiple of one row to another (denoted as R1 + kR2 -> R1) In this case, let's consider the following row operations to achieve an upper triangular form: - R2 - 5R1 → R2 - R3 - 2R1 → R3 - R4 - 6R1 → R4 The new matrix will be: \[ A' = \begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 0 & -6 & -12 & -18 & -24 \\ 0 & -1 & -2 & -3 & -4 \\ 0 & -7 & -14 & -21 & -28 \\ 0 & 2 & 4 & 6 &8 \end{bmatrix} \] Next, we perform the following row operations: - R2 / -6 → R2 - R3 - (-1)R2 → R3 - R4 - (-7)R2 → R4 - R5 + 2R2 → R5 The new matrix will be: \[ A'' = \begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 0 & -1 & -2 & -3 \\ 0 & 0 & -7 & -14 & -21 \\ 0 & 0 & 4 & 8 & 12 \end{bmatrix} \] And then we apply the final row operations: - R3(-1) → R3 - R4 + 7R3 → R4 - R5 - 4R3 → R5 Resulting in the upper triangular matrix: \[ A''' = \begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 1 \end{bmatrix} \]

Step 3: Calculate the determinant

Determine the determinant of the upper triangular matrix A'''. A determinant of an upper triangular matrix is simply the product of the diagonal elements. Using this property, we have: \[ \det(A''') = (1)(1)(1)(1)(1) = 1 \] The determinant of the given matrix A is 1.

Key Concepts

Elementary Row Operations

Elementary row operations are fundamental techniques used in matrix transformations. These operations help simplify matrices and are crucial in methods like Gaussian Elimination. There are three main types of elementary row operations:

• Swapping Rows: Exchanging the positions of two rows. This does not change the determinant’s magnitude but changes its sign.
• Multiplying a Row by a Nonzero Scalar: Every element in a row is multiplied by the same nonzero number. This operation scales the determinant by that scalar.
• Adding or Subtracting Rows: A multiple of one row is added to or subtracted from another row. This operation does not affect the determinant of the matrix. When using these operations, students can transform matrices into a more manageable form, such as an upper triangular matrix, while keeping track of determinant changes. Row operations are thus integral in calculating determinants and solving linear systems.

Upper Triangular Matrix

Understanding an upper triangular matrix is essential in matrix algebra. An upper triangular matrix is one where all elements below the diagonal are zero. Here's why they are helpful:

• Determinant Simplification: The determinant of an upper triangular matrix is simply the product of its diagonal elements. This simplifies calculations drastically.
• Solving Linear Systems: When a matrix is in upper triangular form, you can easily solve systems of linear equations using back-substitution.

The process of transforming a matrix into an upper triangular form is often carried out via Gaussian elimination. This form serves as a stepping stone for further simplifications and calculations in linear algebra. So, when working with matrices, finding an upper triangular form can significantly streamline the process of determinant calculation.

Gaussian Elimination

Gaussian elimination is a systematic method used in linear algebra to solve systems of linear equations. By applying a series of elementary row operations, a matrix is transformed into an easier form: either row echelon or upper triangular.

• Steps Involved: Start with the leftmost pivot and create zeros below each leading coefficient by performing row operations. Continue this process for each subsequent column.
• Creating Triangular Form: The process results in zeros below the diagonal, which is the essence of transforming the matrix into upper triangular form.
• Application: Once in this form, the backward substitution method can be used to find solutions for systems of equations or to calculate determinants efficiently.

Gaussian elimination requires attention to detail with operations to ensure accuracy. It is a robust mechanism for simplifying matrix-based computations and is widely taught to develop problem-solving skills in linear algebra.

Linear Algebra

Linear algebra is a branch of mathematics focusing on vectors, vector spaces, linear transformations, and systems of linear equations. Its concepts are foundational to understanding determinants and matrix operations.

• Matrices and Determinants: Matrices are arrays of numbers representing linear transformations, and their determinants can provide insights into solutions of linear systems.
• Vector Spaces: These are central concepts in linear algebra where linear combinations of vectors form complete spaces.
• Applications: Besides pure mathematics, linear algebra plays significant roles in science and engineering, including computer graphics, signal processing, and optimization tasks.

Through schooling in linear algebra, one gains the skills to handle complex mathematical models. It helps in understanding transformations over vector spaces, solving equation systems, and analyzing matrix properties like determinants.