Problem 15 - Advanced Engineering Mathematics

Question

In Problems 15-18, evaluate the determinant of the given matrix without expanding by cofactors. $$ \mathbf{A}=\left(\begin{array}{rrrr} 6 & 1 & 8 & 10 \\ 0 & \frac{2}{3} & 7 & 2 \\ 0 & 0 & -4 & 9 \\ 0 & 0 & 0 & -5 \end{array}\right) $$

Step-by-Step Solution

Answer

The determinant of the matrix is 80.

Step 1: Identify the Matrix Type

The matrix \( \mathbf{A} \) is an upper triangular matrix. An upper triangular matrix is a square matrix where all the elements below the diagonal are zero.

Step 2: Determine Property of Upper Triangular Matrix

For an upper triangular matrix, the determinant is equal to the product of its diagonal entries. This property simplifies the calculation of the determinant.

Step 3: Calculate the Determinant using Diagonal Elements

The diagonal elements of \( \mathbf{A} \) are 6, \( \frac{2}{3} \), -4, and -5. To find the determinant, multiply these elements: \[\text{Det}(\mathbf{A}) = 6 \times \frac{2}{3} \times (-4) \times (-5)\].

Step 4: Perform the Multiplication

Calculate the product step-by-step: - First, multiply the first two numbers: \[ 6 \times \frac{2}{3} = 4 \]- Next, multiply by the third diagonal element: \[ 4 \times (-4) = -16 \] - Finally, multiply by the fourth diagonal element: \[ -16 \times (-5) = 80 \] The determinant of the matrix \( \mathbf{A} \) is 80.

Key Concepts

Understanding Upper Triangular Matrices

An upper triangular matrix is a specific type of square matrix. In this matrix, every element below the main diagonal is zero. It simply means as you move down any column, starting from the diagonal, you will find only zeros. This structure makes certain matrix operations, like finding the determinant, much easier.
The format can be represented in a general form:

• For any element in row i, column j, if i > j, that element is zero.
• The upper triangular matrix primarily concerns itself with non-zero elements on and above its diagonal.

Understanding the structure of an upper triangular matrix is crucial because it helps in simplifying complex matrix operations. Due to its nature, these matrices appear frequently in algebraic computations and are beneficial for efficient computational calculations.

The Importance of Diagonal Entries

The diagonal entries in a matrix, especially an upper triangular matrix, hold significant importance. These entries sit on the main diagonal that runs from the upper left corner to the lower right corner. In an upper triangular matrix, these diagonal entries are directly used in calculating the determinant, eliminating the need for cofactor expansion.
The characteristic of such a matrix ensures that all impactful information about its determinant is contained solely in its diagonal. Here's what makes diagonal entries special:

• In an upper triangular matrix, the determinant can be calculated by simply multiplying these diagonal values.
• The simplicity of this process is due to the structure, as all other non-diagonal values below the diagonal are zero.

This property of diagonal entries considerably reduces computational complexity, thus saving time and effort.

Key Properties of Matrices

Matrices have various properties that help to perform operations and solve problems efficiently. These properties are especially useful when determining a matrix's behavior or the result of applying certain operations to it. Some core properties include:

• Diagonal Dominance: Where the absolute value of each entry on the diagonal is larger than the sum of the absolute values of the other entries in the same row or column.
• Determinant Calculation: The determinant gives information about the matrix, like whether it is invertible (a non-zero determinant means invertibility).
• Rank: The number of linearly independent rows or columns provides insight into the dimension of the image of the matrix.

Recognizing and using these properties facilitates a deeper understanding of matrices in mathematical and real-world contexts.