Problem 7 - Advanced Engineering Mathematics

Question

In Problems 5-8, suppose $$ \mathbf{A}=\left(\begin{array}{rrrr} 0 & 2 & 4 & 0 \\ 1 & 2 & -2 & 3 \\ 5 & 1 & 0 & -1 \\ 1 & 1 & 1 & 2 \end{array}\right) $$ \text { Evaluate the indicated minor determinant or cofactor. } $$ C_{34} $$

Step-by-Step Solution

Answer

The cofactor \( C_{34} \) is 10.

Step 1: Identify the Cofactor

The cofactor, denoted as \( C_{34} \), refers to the cofactor of the element in the 3rd row, 4th column of the matrix \( \mathbf{A} \). To find \( C_{34} \), we need to calculate the minor, \( M_{34} \), and apply the cofactor formula.

Step 2: Determine the Minor Matrix

To find \( M_{34} \), remove the 3rd row and 4th column from the matrix \( \mathbf{A} \). The resulting 3x3 matrix is: \[\begin{bmatrix}0 & 2 & 4 \1 & 2 & -2 \1 & 1 & 1 \ \end{bmatrix}\]

Step 3: Calculate the Minor Determinant

Calculate the determinant of the 3x3 matrix: \(\begin{vmatrix}0 & 2 & 4 \1 & 2 & -2 \1 & 1 & 1 \end{vmatrix}\)Use the formula for the determinant of a 3x3 matrix:\[= 0 \cdot \begin{vmatrix}2 & -2 \ 1 & 1 \end{vmatrix} - 2 \cdot \begin{vmatrix}1 & -2 \ 1 & 1 \end{vmatrix} + 4 \cdot \begin{vmatrix}1 & 2 \ 1 & 1 \end{vmatrix}\]

Step 4: Solve for the Determinants of 2x2 Submatrices

Find each 2x2 submatrix determinant:1. \( \begin{vmatrix}2 & -2 \ 1 & 1 \end{vmatrix} = (2)(1) - (-2)(1) = 2 + 2 = 4 \) 2. \( \begin{vmatrix}1 & -2 \ 1 & 1 \end{vmatrix} = (1)(1) - (-2)(1) = 1 + 2 = 3 \) 3. \( \begin{vmatrix}1 & 2 \ 1 & 1 \end{vmatrix} = (1)(1) - (2)(1) = 1 - 2 = -1 \)

Step 5: Substitute and Simplify

Substitute the 2x2 determinant results into the 3x3 determinant equation:\[0 \cdot 4 - 2 \cdot 3 + 4 \cdot (-1) = 0 - 6 - 4 = -10\]

Step 6: Calculate the Cofactor

Apply the cofactor formula: \( C_{34} = (-1)^{3+4} \cdot M_{34} = (-1)^7 \cdot (-10) = -(-10) = 10 \)

Key Concepts

Matrix Determinant

A determinant is a special number that can be calculated from a square matrix. It provides insights into the properties of the matrix, such as whether it's invertible. In simple terms, the determinant of a matrix is a scalar value obtained from its elements and can indicate if the matrix represents a singular (non-invertible) transformation or not. To compute a determinant, different formulas are used based on the matrix's size.

• For a 2x2 matrix, there's a straightforward formula.
• For a 3x3 or larger matrix, the process involves finding minors and using cofactor expansion.

Understanding determinants is crucial for solving linear equations, finding inverse matrices, and more.

Minor Matrix

When working with matrices, especially in calculating determinants, the concept of a minor is essential. A minor of a matrix is the determinant of a smaller submatrix that remains after removing one row and one column. For example, to find a minor for a specific element in a matrix, you remove the row and column where that element resides. This results in a smaller matrix, whose determinant gives you the minor. Finding minors is a crucial step in cofactor expansion. Each element's minor is often combined with the cofactor sign factor to contribute to the full determinant calculation of a larger matrix.

3x3 Matrix

A 3x3 matrix is a square matrix with three rows and three columns. Calculating its determinant involves a more complex process than a 2x2 matrix, primarily using cofactor expansion. Here's a refresher on cofactor expansion for a 3x3 matrix:

• Select a row or column to expand across. Often, choosing the row or column with the most zeros simplifies calculations.
• For each element in the selected row/column, multiply the element by its minor's determinant and the cofactor, given by \((-1)^{i+j}\), where \(i\) and \(j\) are the row and column indices of the element.
• Sum the results to obtain the determinant of the 3x3 matrix.

This method is essential for solving various algebraic problems involving matrices.

2x2 Matrix Determinant

Calculating the determinant of a 2x2 matrix is foundational for understanding larger matrices' determinants. The determinant of a 2x2 matrix, with elements \(\begin{pmatrix}a & b \ c & d \end{pmatrix}\), is computed as:

• \(ad - bc\)

This formula is straightforward but fundamental to the cofactor expansion of larger matrices.In problems involving a 3x3 matrix, determining the 2x2 submatrix determinants is necessary when calculating minors. Understanding this simple process helps simplify larger determinant calculations and contributes significantly to the final result.