Problem 14 - Precalculus: Mathematical for Calculus

Question

Evaluate the minor and cofactor using the matrix \(A\) $$A=\left[\begin{array}{rrr} 1 & 0 & \frac{1}{2} \\ -3 & 5 & 2 \\ 0 & 0 & 4 \end{array}\right]$$ $$M_{32}, A_{32}$$

Step-by-Step Solution

Answer

The minor \( M_{32} \) is \(\frac{7}{2}\), and the cofactor \( A_{32} \) is \(-\frac{7}{2}\).

Step 1: Understanding Minors and Cofactors

A minor, denoted by \( M_{ij} \), is calculated by removing the \(i\)-th row and \(j\)-th column from a matrix and finding the determinant of the resulting smaller matrix. A cofactor, denoted by \( A_{ij} \), is the minor times \((-1)^{i+j}\). We need to find \( M_{32} \) and \( A_{32} \), which means we remove the 3rd row and 2nd column for the minor.

Step 2: Remove Elements for \( M_{32} \)

Remove the 3rd row and 2nd column from matrix \( A \). This will give us the following smaller matrix: \[ \left[ \begin{array}{cc} 1 & \frac{1}{2} \ -3 & 2 \end{array} \right] \]

Step 3: Calculate the Minor \( M_{32} \)

To find the determinant of the 2x2 matrix obtained by removing the 3rd row and 2nd column: \[ M_{32} = \det\left( \begin{array}{cc} 1 & \frac{1}{2} \ -3 & 2 \end{array} \right) = (1)(2) - \left(\frac{1}{2}\right)(-3) \]Calculate the determinant: \[ M_{32} = 2 + \frac{3}{2} = \frac{7}{2}. \]

Step 4: Calculate the Cofactor \( A_{32} \)

A cofactor is calculated by multiplying the minor by \((-1)^{i+j}\). For \( A_{32} \), where \(i=3\) and \(j=2\), it becomes: \[ A_{32} = (-1)^{3+2} \times M_{32} = (-1)^5 \times \frac{7}{2} = -\frac{7}{2}. \]

Key Concepts

Determinants

Determinants are a vital concept in linear algebra used to analyze square matrices. A determinant is a special value calculated from a square matrix, which helps in solving linear equations, finding inverse matrices, and determining vector space properties.

For a 2x2 matrix, say \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as:

• \( ad - bc \)

This formula arises from the requirements that determinants should reflect how area or volume scales when a matrix is used to transform space.

Understanding determinants in larger matrices involves breaking them down into smaller 2x2 matrices through minors and cofactors. This skill is fundamental in expanding your matrix algebra toolkit and solving complex problems.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of a matrix, consisting of two rows and two columns. It forms the building block for understanding more complex matrices. In mathematics, the 2x2 matrix is often used to introduce concepts like determinants, inverses, and matrix multiplication because its operations are straightforward while highlighting key algebraic properties.

In the given problem, the 2x2 matrix we derived by removal of row and column is \( \begin{bmatrix} 1 & \frac{1}{2} \ -3 & 2 \end{bmatrix} \). Calculating its determinant follows the formula mentioned earlier:

• Calculate \( (1)(2) = 2 \)
• Compute \( - (\frac{1}{2})(-3) = \frac{3}{2} \)
• Add the results: \( 2 + \frac{3}{2} = \frac{7}{2} \)

2x2 matrices provide a concrete example of how matrices perform transformations, making them crucial in both theoretical and applied settings.

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with matrices and the myriad of operations you can perform on them. It encompasses addition, multiplication, scalar multiplication, transposition, and finding inverses, among others. Mastery of matrix algebra is essential for numerous fields, such as computer science, physics, and economics.

A key aspect of matrix algebra is the concept of minors and cofactors. These are employed in calculating determinants of larger matrices and are intrinsic to finding an inverse of matrices through the adjugate method. In matrix algebra, when evaluating any matrix property, such as the determinant or cofactor as shown in the problem, it's key to remember the foundational rules that matrices follow, such as:

• Only square matrices have determinants.
• Matrix multiplication is not commutative, \( AB eq BA \).

Building a strong understanding of matrix operations can provide the foundation for more advanced topics, like eigenvectors and eigenvalues, thus making matrix algebra a pivotal subject in mathematics.