Problem 4 - Advanced Engineering Mathematics

Question

Suppose $$ \mathbf{A}=\left(\begin{array}{rrr} 2 & 3 & 4 \\ 1 & -1 & 2 \\ -2 & 3 & 5 \end{array}\right) $$ Evaluate the indicated minor determinant or cofactor. $$ C_{22} $$

Step-by-Step Solution

Answer

The cofactor \( C_{22} \) is 18.

Step 1: Locate the Element for Cofactor

The cofactor \( C_{22} \) refers to the position in the second row and second column of matrix \( \mathbf{A} \). The element in this position is \( -1 \).

Step 2: Identify the Minor Matrix

For the cofactor \( C_{22} \), we need to remove the second row and second column from matrix \( \mathbf{A} \). The resulting minor matrix is:\[ \begin{pmatrix} 2 & 4 \ -2 & 5 \end{pmatrix} \]

Step 3: Calculate the Determinant of the Minor

The determinant of the minor matrix \( \begin{pmatrix} 2 & 4 \ -2 & 5 \end{pmatrix} \) is calculated as follows:\[ \text{det} = (2)(5) - (4)(-2) = 10 + 8 = 18 \]

Step 4: Determine the Sign of the Cofactor

The sign of the cofactor is determined by the position \( C_{ij} \) where \( i + j \) is used. In this case, for \( C_{22} \), since \( 2+2 = 4 \), the sign is \( (-1)^4 = +1 \).

Step 5: Multiply the Determinant by the Sign

Multiply the determinant of the minor matrix by the sign factor: \[ C_{22} = (+1) \times 18 = 18 \].

Key Concepts

Minor Determinant

Understanding the concept of a minor determinant is central to calculating matrix determinants. In matrix algebra, a minor of an element is determined by removing the row and column that the element is in. This subtraction effectively reduces the original matrix to a smaller one. For example, in our matrix \( \mathbf{A} \), when calculating the minor for position \( C_{22} \), you remove the second row and second column. This leaves you with a new 2x2 matrix:

• The first row contains the elements \( 2 \) and \( 4 \).
• The second row contains \( -2 \) and \( 5 \).

The minor matrix, therefore, is \( \begin{pmatrix} 2 & 4 \ -2 & 5 \end{pmatrix} \). The determinant of this smaller matrix will be used to calculate the cofactor, highlighting the usefulness of minors. Recognizing and creating these minor matrices is a crucial skill in many matrix algebra processes.

Matrix Algebra

Matrix algebra is a specialized area in mathematics dealing with matrices. Matrices are rectangular arrays of numbers arranged in rows and columns. They are used to solve linear equations, represent complex systems, and perform various transformations. A key operation in matrix algebra is determining the determinant of a matrix, useful for several applications such as calculating inverse matrices or evaluating system properties.
The fundamental operations include:

• Addition and Subtraction: Matrices of the same dimensions can be added or subtracted by adding or subtracting their corresponding elements.
• Multiplication: The number of columns in the first matrix must equal the number of rows in the second for multiplication to be defined.
• Determinant: A special number calculated from a square matrix, essential for various applications including the focus of this exercise.

Understanding these operations is vital for working through more complex matrix problems. Always remember to check the size and arrangement of matrices before performing these operations.

Determinant Calculation

Calculating the determinant is a pivotal aspect of matrix algebra. Determinants give us a scalar value that can provide insights into matrix properties like invertibility, eigenvalues, and orientation. For a 2x2 matrix, the determinant calculation is straightforward:
Given a 2x2 matrix \[ \begin{pmatrix} a & b \ c & d \end{pmatrix}, \]the determinant \( \text{det} \) is computed as:

• \( \text{det} = ad - bc \)

So, for our example, the minor determinant is calculated as \((2)(5) - (4)(-2)\), resulting in 18. This minor's determinant helps determine cofactor values used further in larger determinant evaluations. While simple matrices have an easy calculation, larger matrices require more complex operations, where determinants of smaller matrices (minors) combine to evaluate the whole. Mastering determinant calculations, especially for different matrix sizes, can significantly advance your understanding of matrix algebra.