Question

Find the minors and cofactors of the third row, given \(A=\left[\begin{array}{rrr}9 & 11 & 4 \\ 3 & 2 & 7 \\ 6 & 10 & 4\end{array}\right]\)

Short answer

The minors are 69, 51, -15 and the cofactors are 69, -51, -15.

Step-by-step solution

Identify the Elements of the Third Row

The elements of the third row of matrix \( A \) are 6, 10, and 4.

Calculate the Minor for Element 6

To find the minor of the element 6 in the third row, consider the 2x2 submatrix formed by deleting the third row and the first column:\[\left|\begin{array}{cc}11 & 4 \ 2 & 7\end{array}\right|\]This minor is calculated as:\( 11 \times 7 - 4 \times 2 = 77 - 8 = 69 \).

Calculate the Minor for Element 10

To find the minor of the element 10 in the third row, consider the 2x2 submatrix formed by deleting the third row and the second column:\[\left|\begin{array}{cc}9 & 4 \ 3 & 7\end{array}\right|\]This minor is calculated as:\( 9 \times 7 - 4 \times 3 = 63 - 12 = 51 \).

Calculate the Minor for Element 4

To find the minor of the element 4 in the third row, consider the 2x2 submatrix formed by deleting the third row and the third column:\[\left|\begin{array}{cc}9 & 11 \ 3 & 2\end{array}\right|\]This minor is calculated as:\( 9 \times 2 - 11 \times 3 = 18 - 33 = -15 \).

Calculate the Cofactors

For the third row elements 6, 10, and 4, the cofactors involve applying the sign pattern according to the position \( (-1)^{i+j} \):- For element 6 at (3, 1): \( C_{31} = (-1)^{3+1} \times 69 = 69 \).- For element 10 at (3, 2): \( C_{32} = (-1)^{3+2} \times 51 = -51 \).- For element 4 at (3, 3): \( C_{33} = (-1)^{3+3} \times (-15) = -15 \).

Key concepts

Matrix Theory

Matrix theory is a fascinating branch of mathematics that explores the properties and applications of matrices. A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. In the context of matrix theory, matrices are used to represent linear transformations and solve systems of linear equations.

Understanding matrices involves several key concepts:

• **Matrix Notation**: A matrix is typically denoted by an uppercase letter, such as A. The elements within the matrix are organized into rows and columns.
• **Order of a Matrix**: The size of a matrix is determined by the number of its rows and columns, expressed in the form 'rows x columns'. For example, a 3x3 matrix has three rows and three columns.
• **Operations on Matrices**: Matrices can be added, subtracted, and multiplied. Each of these operations follows specific rules, making matrix calculations unique compared to regular arithmetic.

Matrix theory serves as a foundation for many areas of mathematics and engineering, providing essential tools for solving complex problems.

Linear Algebra

Linear algebra is a central domain in mathematics that deals with vectors, vector spaces, and linear transformations. It is essential for understanding structures and patterns in multi-dimensional spaces. At its core, linear algebra is concerned with vector spaces and the linear mappings between them.

Some of the primary components include:

• **Vectors**: These are objects that have both a magnitude and a direction. They can be thought of as arrows in n-dimensional space.
• **Vector Spaces**: These are collections of vectors that can be added together and multiplied by scalars to produce another vector within the space.
• **Linear Transformations**: These are functions that map vectors to other vectors in such a way that the operations of vector addition and scalar multiplication are preserved.

Linear algebra is widely used in engineering, physics, computer science, and economics due to its ability to model and solve real-world problems effectively.

Determinants

Determinants are a fundamental part of linear algebra, used to determine the solvability of a system of linear equations, among other applications. Finding the determinant helps evaluate the overall scale transformation that a matrix represents when viewed as a linear map.

To understand determinants, consider the following:

• **Square Matrices**: Determinants can only be found for square matrices, i.e., matrices with the same number of rows and columns.
• **Determinant Calculation**: For a 2x2 matrix, the determinant is calculated as the difference in the products of its diagonals. For larger matrices, determinants are calculated using minor matrices and cofactors.
• **Properties of Determinants**: They provide valuable insights about a matrix, such as whether a matrix is invertible. If the determinant of a matrix is zero, then the matrix does not have an inverse, indicating that it represents a degenerate transformation.

Understanding how to compute and utilize determinants is crucial in various mathematical and applied contexts, especially when dealing with linear equations and transformations.