Problem 25 - Precalculus with Limits

Question

In Exercises 25-32, find all (a) minors and (b) cofactors of the matrix. \(\left[ \begin{array}{r} 4 & 5 \\ 3 & -6 \end{array} \right]\)

Step-by-Step Solution

Answer

The minors of the matrix are M11 = -6, M12 = 3, M21 = 5, M22 = 4. The cofactors of the matrix are C11 = 6, C12 = -3, C21 = -5, C22 = 4.

Step 1: Identify the Matrix Elements

The given matrix is a 2x2 matrix with elements: a11 = 4, a12 = 5, a21 = 3 and a22 = -6.

Step 2: Find the Minors

For a 2x2 matrix, the minor of any element is just the other number that is not in the same row or column. So, the minors are: \n- Minor of a11: M11 = a22 = -6 \n- Minor of a12: M12 = a21 = 3 \n- Minor of a21: M21 = a12 = 5 \n- Minor of a22: M22 = a11 = 4

Step 3: Find the Cofactors

The cofactor of any element is found by the formula (-1)^(i+j) * minor. The cofactors are: \n- Cofactor of a11: C11 = (-1)^(1+1) * M11 = 6 \n- Cofactor of a12: C12 = (-1)^(1+2) * M12 = -3 \n- Cofactor of a21: C21 = (-1)^(2+1) * M21 = -5 \n- Cofactor of a22: C22 = (-1)^(2+2) * M22 = 4

Key Concepts

2x2 matrix

A 2x2 matrix is a simple yet fundamental structure in linear algebra, consisting of two rows and two columns. Each position within the grid-like structure is occupied by a number, referred to as the matrix element. Consider the 2x2 matrix from the exercise: \[\begin{bmatrix}4 & 5 \3 & -6\end{bmatrix} \]Here, the number "4" is in the first row and first column, making it element \(a_{11}\), "5" is in the first row and second column as \(a_{12}\), and so on. Such matrices are beneficial because they serve as building blocks for larger matrices. They can also represent a multitude of scenarios and data in fields ranging from physics to economics. In computational processes, dealing with a 2x2 matrix is usually straightforward, enabling quick calculations for determinants, inverses, or transformations.

matrix elements

Matrix elements are the individual numbers that make up a matrix. In a 2x2 matrix, there are four such elements. The exercise presents the elements as:

• \(a_{11} = 4\)
• \(a_{12} = 5\)
• \(a_{21} = 3\)
• \(a_{22} = -6\)

Each element has a specific role. Depending on which element is being focused on, its associated minor and cofactor can be calculated, crucial in processes such as determining a matrix's determinant or adjugate.In context, finding the minor of an element involves identifying the element that remains after removing the current element’s row and column, which is notably just the opposite element in a 2x2 matrix. The cofactor then adjusts this minor with a sign based on its position in the matrix, essentially flipping the signs depending on the matrix space, ensuring calculations align with mathematical conventions.

linear algebra

Linear algebra is a core branch of mathematics focused on vector spaces and linear equations. It deals with concepts like matrices, determinants, and systems of linear equations, making it integral in various scientific and engineering fields. In linear algebra, a matrix like our 2x2 example is used to solve systems of linear equations. This is because a matrix can compactly represent these equations, allowing them to be manipulated and solved systematically. For example, to find the minors and cofactors of its elements, we rely on specific steps and conventions inherent in linear algebra. Understanding these concepts aids in further operations like determining the matrix inverse or performing eigenvalue computations, which have real-world applications, from computer graphics transformations to economic modeling. The principles of linear algebra are foundational, leading to broader and deeper mathematical concepts such as vector spaces and tensor analysis.