Problem 20 - Advanced Engineering Mathematics

Question

Evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{lll} \frac{1}{4} & 6 & 0 \\ \frac{1}{3} & 8 & 0 \\ \frac{1}{2} & 9 & 0 \end{array}\right) $$

Step-by-Step Solution

Answer

The determinant of the matrix is 0.

Step 1: Understanding the matrix

We are given a 3x3 matrix: \[A = \begin{pmatrix}\frac{1}{4} & 6 & 0 \\frac{1}{3} & 8 & 0 \\frac{1}{2} & 9 & 0 \end{pmatrix}\]. We are tasked with finding its determinant using cofactor expansion.

Step 2: Choose the row or column for expansion

Notice that the third column of the matrix contains all zeros. Cofactor expansion along a column or row with zeros simplifies the computation, as any term involving those zeros will not contribute to the determinant.

Step 3: Apply cofactor expansion

We will perform cofactor expansion along the third column. The formula for the determinant using the third column is:\[\text{det}(A) = a_{13}C_{13} + a_{23}C_{23} + a_{33}C_{33}\]where each entry \(a_{ij}\) from the third column is zero, i.e., \(a_{13}=0, a_{23}=0, a_{33}=0\).

Step 4: Simplify

Since each term in the cofactor expansion involves multiplying by zero, every part of the expansion becomes zero:\[\text{det}(A) = 0 \cdot C_{13} + 0 \cdot C_{23} + 0 \cdot C_{33} = 0\]Thus, the determinant of the matrix is zero.

Key Concepts

Understanding Cofactor Expansion

Cofactor expansion is a method used to calculate the determinant of a square matrix, which can be particularly useful for matrices of size 3x3 as in this exercise. This method involves breaking down the determinant calculation into smaller, more manageable parts, each of which is a product of a matrix element and its corresponding cofactor.

When using cofactor expansion, we choose a specific row or column to expand along. This choice influences the calculation, especially if the chosen row or column contains zeros, as any terms involving a zero will not affect the overall determinant. Each element of the chosen row or column is multiplied by its cofactor, a smaller determinant which excludes the row and column of the chosen element.

The cofactor itself is calculated as follows: for an element at position \(a_{ij}\) in the matrix, its cofactor \(C_{ij}\) is given by

• The determinant of the (n-1) x (n-1) submatrix formed by removing the \(i^{th}\) row and \(j^{th}\) column from the original matrix.
• It is then multiplied by \((-1)^{i+j}\) to determine its sign.

This alternating sign ensures correct orientation in the determinant's calculation.

Characteristics of a 3x3 Matrix

A 3x3 matrix consists of three rows and three columns, presenting a square configuration that is a common matrix size in linear algebra. Working with 3x3 matrices, particularly in the context of cofactor expansion and finding determinants, serves as a foundational skill in understanding more complex matrix operations.

Key characteristics of a 3x3 matrix include:

• It has exactly nine elements arranged in rows and columns.
• The determinant of a 3x3 matrix can often be simplified if certain elements, such as zeros, are present. As seen in the exercise, columns with zeros make cofactor expansion straightforward and quick.
• The structure and arrangement directly impact operations like matrix multiplication, inversion, and calculating its determinant.

In linear algebra, the 3x3 matrix serves as a versatile tool, applicable in various computations, including solving systems of linear equations, transformations, and more.

Role of Determinants in Linear Algebra

Determinants play a crucial role in the field of linear algebra, providing insightful characteristics of square matrices that influence many mathematical applications. The determinant of a matrix, such as a 3x3 matrix, is a scalar value that gives information about the matrix properties, including invertibility, volume scaling in transformations, and more.

Here are some important aspects of determinants:

• A non-zero determinant indicates that the matrix is invertible, suggesting unique solutions in linear systems.
• Conversely, a zero determinant, as seen in the exercise, means the matrix is singular, implying no unique solutions exist or the matrix cannot be inverted.
• Determinants are also used to determine the orientation and scaling factors of linear transformations represented by matrices.

Understanding determinants leads to a deeper comprehension of concepts such as matrix rank, eigenvalues, and eigenvectors, all of which are integral parts of linear algebra. This knowledge is fundamental when analyzing how linear systems behave and predicting their solutions.