Problem 41 - Discrete Mathematics with Applications

Question

Let \(A, B,\) and \(C\) be square matrices of order \(2 .\) Prove each. $$(A B)^{\mathrm{T}}=B^{\mathrm{T}} A^{\mathrm{T}}$$

Step-by-Step Solution

Answer

Given square matrices A, B, and C of order 2, we want to prove that \((A B)^{T}=B^{T} A^{T}\). 1. Define the transpose of a matrix: \(A_{ji}^T = A_{ij}\). 2. Define matrix multiplication: \((A B)_{ij} = \sum_{k=1}^{2} A_{ik} B_{kj}\). 3. Compute the transpose of the product: \(((AB)^T)_{ij} = \sum_{k=1}^{2} A_{jk} B_{ki}\). 4. Compute the product of individual transposes: \((B^T A^T)_{ij} = \sum_{k=1}^{2} B_{ki} A_{jk}\). Comparing the expressions from Steps 3 and 4, we see they are equal: \[((AB)^T)_{ij} = (B^T A^T)_{ij}\] Thus, we have proved that \((A B)^{T}=B^{T} A^{T}\).

Step 1: Definition of Transpose

Recall that the transpose of a matrix is the matrix obtained by interchanging its rows and columns. Given matrix A, its transpose is denoted as A^T, and the elements of the transpose matrix are defined as: \[A_{ji}^T = A_{ij}\]

Step 2: Definition of Matrix Multiplication

Recall that the product of two matrices (A and B) is defined as: \[(A B)_{ij} = \sum_{k=1}^{2} A_{ik} B_{kj}\] Here, i and j are the row and column indices of the product matrix.

Step 3: Compute the Transpose of the Product

To compute the transpose of the product matrix (AB), let's consider the (i,j) element of its transpose: \[((AB)^T)_{ij}\] Using the definition of transpose from Step 1: \[((AB)^T)_{ij} = (AB)_{ji}\] Using the definition of matrix multiplication from Step 2: \[((AB)^T)_{ij} = \sum_{k=1}^{2} A_{jk} B_{ki}\]

Step 4: Compute the Product of Individual Transposes

Now, let's compute the product B^T A^T, and its (i,j) element: \[(B^T A^T)_{ij}\] Using the definition of matrix multiplication from Step 2: \[(B^T A^T)_{ij} = \sum_{k=1}^{2} (B^T)_{ik} (A^T)_{kj}\] Using the definition of transpose from Step 1: \[(B^T A^T)_{ij} = \sum_{k=1}^{2} B_{ki} A_{jk}\]

Step 5: Comparing Both Sides

Now, let's examine the two expressions we derived in Steps 3 and 4: \[((AB)^T)_{ij} = \sum_{k=1}^{2} A_{jk} B_{ki}\] \[(B^T A^T)_{ij} = \sum_{k=1}^{2} B_{ki} A_{jk}\] Since both expressions are equal, we have: \[((AB)^T)_{ij} = (B^T A^T)_{ij}\] Thus, we have proved that: \[(A B)^{T}=B^{T} A^{T}\]

Key Concepts

Matrix Multiplication

Matrix multiplication is an essential operation in linear algebra involving two matrices, resulting in a new matrix. The key process for multiplying matrices involves taking the rows of the first matrix and the columns of the second. Here's a simple breakdown to aid your understanding:

• The number of columns in the first matrix must equal the number of rows in the second matrix.
• Each element of the resulting matrix is computed as the sum of the products of corresponding elements from the rows of the first matrix and the columns of the second matrix.

Let's consider two 2x2 matrices, \( A \) and \( B \). When multiplying these, each element of the resulting matrix \( C = AB \) is calculated as follows:\[C_{ij} = \sum_{k=1}^{2} A_{ik} B_{kj}\]By understanding this process, you can tackle matrix multiplication tasks confidently. It's crucial because this operation is widely used in applications such as computer graphics and solving systems of linear equations.

Square Matrices

Square matrices are matrices that have the same number of rows and columns. For instance, a matrix of order 2 is a 2x2 square matrix. These types of matrices are particularly interesting because they provide certain properties not seen in non-square matrices.

Some key characteristics of square matrices include:

• They can have a determinant, which is a special number that can be calculated from its elements.
• Square matrices are often involved in processes like finding eigenvalues and eigenvectors.
• Only square matrices can have an inverse, which is a matrix that, when multiplied with the original, yields the identity matrix.

Identity and Transpose:Square matrices often appear in expressions involving the transpose or identity. The identity matrix is a square matrix with ones on the diagonal and zeroes elsewhere, acting like the number 1 in multiplication. For a matrix \( A \), the transpose, denoted \( A^T \), is another square matrix, resulting from switching elements across the diagonal.

Order of Matrices

The order of a matrix provides crucial information about its structure and operations that can be performed on it. Here, the order of a matrix refers to its dimensions, given as "m by n" (\(m \times n\)), where \(m\) is the number of rows and \(n\) is the number of columns.

Understanding the order is vital because:

• It dictates compatibility in operations like addition, where only matrices of the same order can be added or subtracted.
• It determines whether matrices can be multiplied. For two matrices to be multiplied, the number of columns in the first matrix must match the number of rows in the second.
• The order tells you about the potential for special properties like being a square matrix.

When working with matrices, always consider their order to understand what operations you can perform and what properties they may possess. This ensures accurate computation and meaningful results in mathematical and practical applications.