If Matrix A = \begin{pmatrix} 4 & -2 \\ 5 & -3 \end{pmatrix}, Prove That The Determinant Of A Transpose Equals The Determinant Of A, Or |A'| = |A|.

Introduction to Determinants and Transpose of a Matrix

In linear algebra, the determinant of a matrix is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. The determinant of a matrix A is denoted as |A| or det(A). It plays a crucial role in various applications, such as solving systems of linear equations, finding eigenvalues, and determining the invertibility of a matrix. Understanding the properties of determinants is fundamental for advanced topics in mathematics, physics, and engineering. In this article, we delve into one of the key properties of determinants, specifically focusing on the relationship between the determinant of a matrix and the determinant of its transpose. The transpose of a matrix, denoted as A′, is formed by interchanging the rows and columns of the original matrix A. This operation might seem simple, but it has significant implications when dealing with determinants. The property we aim to explore and prove here is that the determinant of the transpose of a matrix is equal to the determinant of the original matrix itself, i.e., |A′| = |A|. This equality is not just a mathematical curiosity; it has practical applications in simplifying calculations and proving other theorems in linear algebra. To illustrate this property, we will consider a specific 2x2 matrix A and compute both |A| and |A′| to verify their equality. This hands-on approach will provide a clear and intuitive understanding of the concept, making it easier to grasp the more general proof and applications of this property. Furthermore, we will discuss why this property holds true and its broader implications in the context of matrix operations and linear transformations. By understanding the relationship between the determinant of a matrix and its transpose, we can gain deeper insights into the behavior of matrices and their applications in various fields.

Problem Statement: Verifying |A'| = |A|

Our primary objective is to demonstrate that for a given matrix A, the determinant of its transpose (A′) is equal to the determinant of the original matrix (A). Specifically, we are given the matrix A = \begin{pmatrix} 4 & -2 \ 5 & -3 \end{pmatrix} and we need to show that |A′| = |A|. This problem serves as a practical exercise in understanding the properties of determinants and matrix transposes. To achieve this, we will first compute the transpose of the matrix A, which involves interchanging its rows and columns. Once we have A′, we will calculate its determinant. Subsequently, we will compute the determinant of the original matrix A. By comparing the two results, we aim to verify that they are indeed equal, thus validating the property |A′| = |A|. This process not only reinforces our understanding of matrix operations but also provides a concrete example of an important theorem in linear algebra. The significance of this property extends beyond mere computation; it highlights a fundamental characteristic of determinants that is useful in various theoretical and applied contexts. For instance, this property can simplify calculations in more complex problems and is often used in proofs involving determinants. Moreover, understanding this relationship helps in grasping the broader implications of matrix operations on the properties of linear transformations. The exercise of computing determinants and transposes provides valuable practice in matrix algebra, which is essential for anyone working with linear systems, vector spaces, and other related concepts. By systematically working through this problem, we will not only confirm the given property but also enhance our skills in matrix manipulation and determinant calculation. This will lay a solid foundation for tackling more advanced topics in linear algebra and its applications.

Step-by-Step Solution

To show that |A′| = |A| for the given matrix A = \begin{pmatrix} 4 & -2 \ 5 & -3 \end{pmatrix}, we will follow these steps:

1. Find the Transpose of Matrix A (A')

The transpose of a matrix is obtained by interchanging its rows and columns. For matrix A, the first row becomes the first column, and the second row becomes the second column. Thus, the transpose of A, denoted as A′, is:

A′ = \begin{pmatrix} 4 & 5 \ -2 & -3 \end{pmatrix}

2. Calculate the Determinant of A (|A|)

The determinant of a 2x2 matrix \begin{pmatrix} a & b \ c & d \end{pmatrix} is calculated as ad - bc. Therefore, the determinant of A is:

|A| = (4 × -3) - (-2 × 5) = -12 - (-10) = -12 + 10 = -2

3. Calculate the Determinant of A' (|A'|)

Similarly, we calculate the determinant of A′ using the same formula:

|A′| = (4 × -3) - (5 × -2) = -12 - (-10) = -12 + 10 = -2

4. Compare |A'| and |A|

Comparing the two results, we find that:

|A′| = -2

|A| = -2

Therefore, |A′| = |A|.

Detailed Explanation of the Steps

Finding the Transpose (A')

The first step in our solution involves finding the transpose of the matrix A. The transpose of a matrix is a fundamental operation in linear algebra, and it is essential for various matrix manipulations and proofs. To find the transpose, we simply interchange the rows and columns of the original matrix. This means that the elements in the first row of A become the elements in the first column of A′, and similarly, the elements in the second row of A become the elements in the second column of A′. For the given matrix A = \begin{pmatrix} 4 & -2 \ 5 & -3 \end{pmatrix}, this process is straightforward. The first row (4, -2) becomes the first column, and the second row (5, -3) becomes the second column. This results in the transpose matrix A′ = \begin{pmatrix} 4 & 5 \ -2 & -3 \end{pmatrix}. Understanding how to find the transpose is crucial as it not only helps in solving problems like this but also plays a significant role in more advanced concepts such as eigenvalues and eigenvectors, orthogonal matrices, and symmetric matrices. The transpose operation is also essential in various applications, including data analysis, computer graphics, and physics, where matrices are used to represent transformations and systems of equations. By mastering this basic operation, we can build a solid foundation for more complex matrix operations and their applications. The transpose operation is not just a mechanical process; it has deep connections to the underlying linear transformations represented by the matrices. For example, the transpose of a matrix can be related to the adjoint of a linear transformation, which provides insights into the geometric interpretation of the transformation.

Calculating the Determinant of A (|A|)

After finding the transpose, the next step is to calculate the determinant of the original matrix A. The determinant is a scalar value that can be computed from the elements of a square matrix and provides valuable information about the matrix, such as its invertibility and the volume scaling factor of the corresponding linear transformation. For a 2x2 matrix, the determinant is calculated using a simple formula: if A = \begin{pmatrix} a & b \ c & d \end{pmatrix}, then |A| = ad - bc. This formula is easy to apply and is fundamental for understanding the properties of determinants. In our case, the matrix A = \begin{pmatrix} 4 & -2 \ 5 & -3 \end{pmatrix}, so we identify a = 4, b = -2, c = 5, and d = -3. Plugging these values into the formula, we get |A| = (4 × -3) - (-2 × 5) = -12 - (-10) = -12 + 10 = -2. The result, -2, is the determinant of matrix A. The determinant being a non-zero value indicates that the matrix A is invertible, meaning there exists a matrix A⁻¹ such that A A⁻¹ = I, where I is the identity matrix. The determinant also provides insights into the geometric interpretation of the linear transformation represented by A. In this case, the negative determinant suggests that the transformation involves a reflection, and the absolute value of the determinant, 2, indicates the scaling factor of the area under the transformation. Understanding the determinant is crucial for various applications, including solving systems of linear equations, finding eigenvalues and eigenvectors, and determining the stability of systems in physics and engineering. The determinant is not just a numerical value; it is a powerful tool for analyzing and understanding the properties of matrices and the linear transformations they represent.

Calculating the Determinant of A' (|A'|)

Following the calculation of the determinant of the original matrix A, we now need to compute the determinant of its transpose, A′. This step is crucial for verifying the property |A′| = |A|. We use the same formula for calculating the determinant of a 2x2 matrix: if A′ = \begin{pmatrix} a & b \ c & d \end{pmatrix}, then |A′| = ad - bc. The transpose matrix A′ we found earlier is A′ = \begin{pmatrix} 4 & 5 \ -2 & -3 \end{pmatrix}. Identifying the elements, we have a = 4, b = 5, c = -2, and d = -3. Plugging these values into the formula, we get |A′| = (4 × -3) - (5 × -2) = -12 - (-10) = -12 + 10 = -2. The result, -2, is the determinant of the transpose matrix A′. This calculation is a direct application of the determinant formula and reinforces the process of computing determinants for 2x2 matrices. The determinant of the transpose provides valuable information about the transformation represented by A′. In this case, the determinant being -2 indicates that the transformation involves a reflection and scales the area by a factor of 2, similar to the original matrix A. The fact that the determinant of A′ is the same as the determinant of A is not a coincidence; it is a fundamental property of determinants that holds true for all square matrices. This property is significant because it simplifies various calculations and proofs in linear algebra. For instance, when dealing with complex matrix operations, knowing that |A′| = |A| can reduce the computational burden and provide a shortcut in problem-solving. Moreover, this property is often used in theoretical arguments and proofs related to determinants and matrix transformations. Understanding how to calculate the determinant of a transpose and recognizing its equality to the determinant of the original matrix is essential for a deeper comprehension of linear algebra concepts and their applications.

Comparing |A'| and |A|

After independently calculating the determinants of both the original matrix A and its transpose A′, the final step is to compare the results. This comparison is the crux of the problem, as it allows us to verify the property that the determinant of a matrix is equal to the determinant of its transpose. From our previous calculations, we found that |A| = -2 and |A′| = -2. By simply observing these two values, it is evident that they are equal. This confirms the property |A′| = |A| for the given matrix A. The equality of these determinants is not just a numerical coincidence; it is a fundamental property in linear algebra that holds true for all square matrices. This property has significant implications in various theoretical and practical applications. For example, in the context of linear transformations, it implies that the scaling factor of the area (or volume in higher dimensions) remains the same whether we apply the original transformation or its transpose. This is because the determinant represents the scaling factor of the transformation. Furthermore, this property simplifies many calculations and proofs involving determinants. For instance, if we need to calculate the determinant of a complex matrix expression involving transposes, knowing that |A′| = |A| can help us simplify the expression and reduce the computational effort. The fact that the determinant is invariant under transposition also has connections to other concepts in linear algebra, such as eigenvalues and eigenvectors. In summary, the comparison of |A| and |A′| is not just a final check in our solution; it is a validation of an important property that enriches our understanding of determinants and their role in matrix algebra and linear transformations. This property is a cornerstone in the broader theory of matrices and their applications.

Conclusion: The Significance of |A'| = |A|

In conclusion, we have successfully demonstrated that for the given matrix A = \begin{pmatrix} 4 & -2 \ 5 & -3 \end{pmatrix}, the determinant of its transpose, A′, is equal to the determinant of A. This verification not only provides a concrete example of the property |A′| = |A| but also underscores the broader significance of this property in linear algebra. The fact that the determinant remains unchanged under transposition is a fundamental characteristic of determinants and has far-reaching implications. This property is not merely a mathematical curiosity; it simplifies various calculations and proofs, and it offers insights into the nature of linear transformations. For instance, it implies that the scaling factor of a transformation (represented by the determinant) is preserved even when the transformation is transposed. This has practical relevance in fields such as computer graphics, where transformations are used to manipulate objects in space, and in physics, where transformations are used to describe changes in coordinate systems. Moreover, the property |A′| = |A| is often used as a building block in more advanced theorems and concepts in linear algebra. It plays a role in the study of eigenvalues, eigenvectors, orthogonal matrices, and the invertibility of matrices. Understanding this property helps in developing a deeper understanding of the structure and behavior of matrices and their applications. The ability to manipulate determinants and transposes effectively is a valuable skill for anyone working with linear systems, vector spaces, and matrix algebra. It allows for more efficient problem-solving and a more intuitive grasp of the underlying mathematical principles. In essence, the property |A′| = |A| is a testament to the elegant and interconnected nature of linear algebra, where seemingly simple operations and properties can lead to profound insights and applications.