Problem 15 - Algebra and Trigonometry

Question

Evaluate the minor and cofactor using the matrix \(A\) $$ A=\left[\begin{array}{rrr}{1} & {0} & {\frac{1}{2}} \\ {-3} & {5} & {2} \\\ {0} & {0} & {4}\end{array}\right] $$ $$ M_{12}, A_{12} $$

Step-by-Step Solution

Answer

The minor \( M_{12} \) is -12 and the cofactor \( A_{12} \) is 12.

Step 1: Identify the Minor

The minor, denoted as \( M_{12} \), is obtained by deleting the first row and second column from the matrix \( A \). This results in the submatrix:\[\begin{bmatrix}-3 & 2 \0 & 4\end{bmatrix}\]

Step 2: Calculate the Determinant of the Submatrix

To find \( M_{12} \), calculate the determinant of the 2x2 submatrix obtained in Step 1:\[M_{12} = \begin{vmatrix} -3 & 2 \ 0 & 4 \end{vmatrix}\]The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated as \( ad - bc \). Applying this formula gives:\[M_{12} = (-3)(4) - (2)(0) = -12\]

Step 3: Identify the Cofactor

The cofactor, denoted as \( A_{12} \), is calculated using the formula:\[A_{ij} = (-1)^{i+j} imes M_{ij}\]For \( i = 1 \) and \( j = 2 \), substituting the values gives:\[A_{12} = (-1)^{1+2} imes M_{12} = (-1)^3 imes (-12)\]

Step 4: Calculate the Cofactor

Substituting in the sign change from Step 3:\[A_{12} = -1 imes (-12) = 12\]

Key Concepts

Minor of a Matrix

To understand the concept of a minor in matrix algebra, let's first define a minor. A minor is the determinant of a smaller square matrix, called a submatrix, which you obtain by removing one or more rows and columns from a larger square matrix. In our exercise, we encountered the 3x3 matrix \( A \). We needed to determine the minor \( M_{12} \). This involves removing the 1st row and the 2nd column of matrix \( A \).
This process leaves us with the following submatrix:

• Row Removal: Achieved by skipping the entire first row
• Column Removal: Achieved by disregarding the entire second column

Once we have the submatrix \( \begin{bmatrix}-3 & 2 \ 0 & 4 \end{bmatrix} \), we can calculate its determinant to find \( M_{12} \). The result of this calculation gives us the minor of the original matrix.

Cofactor of a Matrix

Once you understand what a minor is, the next step is to delve into the concept of a cofactor. A cofactor is closely related to a minor but also considers the position of the element. For a given element in matrix \( A \), the cofactor \( A_{ij} \) is computed by

• First, determining the minor associated with element \( A_{ij} \)
• Then, multiplying this minor by \((-1)^{i+j}\)

This multiplication introduces a sign based on the element's position within the matrix. In our example, for the element in the first row and second column (\( i = 1, j = 2 \)), we computed \( A_{12} = (-1)^{1+2} \times M_{12} \). The negative sign from \((-1)^3\) changes the sign of the minor \( -12 \) to give \( 12 \). Thus, the cofactor reflects both the size and position of the original matrix element.

2x2 Determinant Calculation

Calculating the determinant of a 2x2 matrix is one of the foundational tasks in matrix algebra. It serves as the basis for finding minors and cofactors. For any 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated using the formula \( ad - bc \).
In the exercise, once we isolated the submatrix obtained from matrix \( A \), we performed this basic operation on \( \begin{bmatrix}-3 & 2 \ 0 & 4 \end{bmatrix} \). Here, we have:

• \( a = -3 \), \( b = 2 \)
• \( c = 0 \), \( d = 4 \)
• Determinant: \((-3 \times 4) - (2 \times 0) = -12 \)

Thus, the determinant \( -12 \) indicates the minor we needed. This method is both simple and effective for quick calculations in matrix operations.

Matrix Algebra

Matrix algebra encompasses a wide range of operations and theories. These operations include addition, subtraction, multiplication, as well as finding determinants and inverses. The study of these matrices explores numerical and algebraic relationships, crucial for solving complex math and physics problems.
In the context of our exercise:

• We worked on evaluating a minor and cofactor, two fundamental components
• These elements become vital when discussing topics like adjoints and inverses

By improving your comfort with single and basic operations like finding minors and cofactors, you lay the groundwork for tackling more advanced matrix-based problems shortly. Understanding these basics in matrix algebra will significantly enhance your ability to manipulate and apply matrices in higher-level math courses.

Algebraic Expressions

Dealing with matrices inherently involves handling algebraic expressions. This is because, in many cases, the elements involved are themselves expressions rather than just numerical values. Algebraic manipulation becomes essential, especially when you deal with variables as elements within your matrix.
Though our specific exercise was numerical, it can be extended to algebraic scenarios where elements like \( x \) or \( y \) replace constants:

• Understanding how to handle the symbols and operations involved is critical
• This builds a scaffolding for recognizing patterns and solving equations

As you develop these skills, remember that algebraic expressions are the foundation of manipulating matrices and other mathematical structures, helping you in simplifying expressions and solving more involved problems.