Problem 30 - Advanced Engineering Mathematics

Question

In Problems 29 and 30, evaluate the determinant of the given matrix by inspection. $$ \left(\begin{array}{rrrr} -3 & 0 & 0 & 0 \\ 4 & 6 & 0 & 0 \\ 1 & 3 & 9 & 0 \\ 6 & 4 & 2 & 1 \end{array}\right) $$

Step-by-Step Solution

Answer

The determinant is -162.

Step 1: Understand the Matrix Type

The given matrix is a 4x4 matrix. We need to observe if it has any special properties, like being an upper or lower triangular matrix, to simplify the determinant calculation.

Step 2: Identify the Triangular Nature

Observe that the matrix is a lower triangular matrix, as all the elements above the main diagonal are zero. Lower triangular matrices have all non-zero elements on or below the main diagonal.

Step 3: Apply the Determinant Formula for Triangular Matrices

For a triangular matrix, whether lower or upper, the determinant is the product of all diagonal elements. In this matrix, the diagonal elements are \(-3, 6, 9, 1\).

Step 4: Calculate the Determinant

Multiply the diagonal elements: \(-3 \times 6 \times 9 \times 1\). Calculate each step clearly:1. \(-3 \times 6 = -18\)2. \(-18 \times 9 = -162\)3. \(-162 \times 1 = -162\)Thus, the determinant is \(-162\).

Key Concepts

Understanding Triangular Matrices

In linear algebra, a key concept that frequently appears is the idea of a **triangular matrix**. A triangular matrix is a special type of square matrix where all the entries are either below or above the main diagonal, significantly simplifying many mathematical operations, especially the calculation of determinants.
A triangular matrix can be categorized into two types:

• **Lower Triangular Matrix**: All the elements above the main diagonal are zero. The matrix in the exercise is a lower triangular matrix since its upper portion, above the main diagonal, consists only of zero values.
• **Upper Triangular Matrix**: All the elements below the main diagonal are zero.

These matrices are widely used because they simplify complex operations, making them powerful tools when solving equations or evaluating determinants. Knowing that you have a triangular matrix allows you to efficiently compute the determinant simply by multiplying the diagonal elements.

Role of Diagonal Elements

The **diagonal elements** of a matrix are the elements that lie on the diagonal that runs from the top left to the bottom right. In mathematical terms, for a square matrix, these are the elements where the row and column indices are the same (i.e., positions \((i,i)\)). These diagonal elements play a crucial role when evaluating the determinant of triangular matrices.
Why are diagonal elements so important?

• In triangular matrices, the determinant can be found by a simple multiplication of these diagonal elements due to their unique structural properties.
• For the matrix given in the exercise, the diagonal elements are \(-3, 6, 9,\) and \(1\). Simply multiplying these gives the determinant of the matrix, which is \(-3 \times 6 \times 9 \times 1 = -162\).
• This rule is specific to triangular matrices because the zero elements in triangular positions do not affect the determinant directly.

Understanding how diagonal elements affect a matrix's structure and determinant can make solving these problems fast and intuitive.

Essential Ideas in Linear Algebra

**Linear algebra** is a branch of mathematics that deals with vectors, matrices, and linear transformations. It's foundational for many areas of mathematics and applied sciences. A few core concepts crucial for understanding the problem include:

• **Matrix and Determinant:** A matrix is a rectangular array of numbers. The determinant is a special number that can be calculated from its elements. For square matrices, determinants are primarily used to solve linear systems and analyze matrix properties, such as invertibility.
• **Matrix Types:** Recognizing the structure, such as whether a matrix is lower triangular, upper triangular, or diagonal, helps in simplifying calculations, as seen in the exercise.
• **Operations Simplification:** Knowing specific properties of matrices, including triangular forms, can drastically simplify operations—as with our determinant calculation in the exercise.

Linear algebra is powerful because it provides the tools to solve complex system equations simply and understandably. Recognizing patterns, such as when dealing with triangular matrices, is a skill that can save you time and computational effort.