Problem 32 - Advanced Engineering Mathematics

Question

In Problems 31-34, suppose \(\mathbf{A}=\left(\begin{array}{rr}2 & 4 \\ -3 & 2\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rr}4 & 10 \\ 2 & 5\end{array}\right)\). Verify the given property by computing the left and right members of the given equality. $$ (\mathbf{A}+\mathbf{B})^{T}=\mathbf{A}^{T}+\mathbf{B}^{T} $$

Step-by-Step Solution

Answer

Yes, the equality \((\mathbf{A} + \mathbf{B})^{T} = \mathbf{A}^{T} + \mathbf{B}^{T}\) is verified.

Step 1: Calculate \(\mathbf{A} + \mathbf{B}\)

Add matrix \(\mathbf{A}\) and matrix \(\mathbf{B}\) element-wise.\[\mathbf{A} = \begin{pmatrix} 2 & 4 \ -3 & 2 \end{pmatrix}\quad\text{and}\quad\mathbf{B} = \begin{pmatrix} 4 & 10 \ 2 & 5 \end{pmatrix}\]So,\[\mathbf{A} + \mathbf{B} = \begin{pmatrix} 2+4 & 4+10 \ -3+2 & 2+5 \end{pmatrix} = \begin{pmatrix} 6 & 14 \ -1 & 7 \end{pmatrix}\]

Step 2: Transpose \((\mathbf{A} + \mathbf{B})\)

Transpose the resulting matrix from Step 1 by swapping its rows and columns.\[(\mathbf{A} + \mathbf{B})^{T} = \begin{pmatrix} 6 & 14 \ -1 & 7 \end{pmatrix}^{T} = \begin{pmatrix} 6 & -1 \ 14 & 7 \end{pmatrix}\]

Step 3: Transpose \(\mathbf{A}\)

Transpose matrix \(\mathbf{A}\).\[\mathbf{A}^{T} = \begin{pmatrix} 2 & 4 \ -3 & 2 \end{pmatrix}^{T} = \begin{pmatrix} 2 & -3 \ 4 & 2 \end{pmatrix}\]

Step 4: Transpose \(\mathbf{B}\)

Transpose matrix \(\mathbf{B}\).\[\mathbf{B}^{T} = \begin{pmatrix} 4 & 10 \ 2 & 5 \end{pmatrix}^{T} = \begin{pmatrix} 4 & 2 \ 10 & 5 \end{pmatrix}\]

Step 5: Calculate \(\mathbf{A}^{T} + \mathbf{B}^{T}\)

Add the transposed matrices from Step 3 and Step 4 element-wise.\[\mathbf{A}^{T} + \mathbf{B}^{T} = \begin{pmatrix} 2 & -3 \ 4 & 2 \end{pmatrix} + \begin{pmatrix} 4 & 2 \ 10 & 5 \end{pmatrix} = \begin{pmatrix} 2+4 & -3+2 \ 4+10 & 2+5 \end{pmatrix} = \begin{pmatrix} 6 & -1 \ 14 & 7 \end{pmatrix}\]

Step 6: Verify the Property

Compare the results from Step 2 and Step 5 to check the equality. Both results are \(\begin{pmatrix} 6 & -1 \ 14 & 7 \end{pmatrix}\), confirming \((\mathbf{A} + \mathbf{B})^{T} = \mathbf{A}^{T} + \mathbf{B}^{T}\).

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra, essential when working with matrices. It's relatively straightforward and involves adding matrices of the same dimensions element-wise. For example, if you have two matrices \( \mathbf{A} \) and \( \mathbf{B} \), each with elements like \( \mathbf{A} = \begin{pmatrix} 2 & 4 \ -3 & 2 \end{pmatrix} \) and \( \mathbf{B} = \begin{pmatrix} 4 & 10 \ 2 & 5 \end{pmatrix} \), the sum \( \mathbf{A} + \mathbf{B} \) is computed as follows:

• Add corresponding elements: \( 2+4 \), \( 4+10 \)
• Continue for each element: \( -3+2 \), \( 2+5 \)

This gives us a new matrix: \( \mathbf{A} + \mathbf{B} = \begin{pmatrix} 6 & 14 \ -1 & 7 \end{pmatrix} \).
Matrix addition must meet certain conditions such as both matrices needing to have the same size. You can't add matrices of different dimensions directly because the operation must match each element by its position in both matrices. This operation is both commutative and associative, meaning you can switch the order of matrices or group them differently when adding.

Matrix Properties

Understanding the properties of matrices is crucial, especially when dealing with operations like transposition and addition. When looking at the transposition property, there is a particular rule, termed the transpose of a sum, which you'll often encounter: \( (\mathbf{A} + \mathbf{B})^{T} = \mathbf{A}^{T} + \mathbf{B}^{T} \).
Transposition involves flipping a matrix over its diagonal, swapping the row and column indices of each element. Therefore, when you transpose a matrix, the rows become columns and the columns become rows. Applying this to matrix addition, you can see that the transpose of the sum of two matrices will equal the sum of their transposes:

• First, perform addition: \( \mathbf{A} + \mathbf{B} = \begin{pmatrix} 6 & 14 \ -1 & 7 \end{pmatrix} \)
• Then transpose the result: \( (\mathbf{A} + \mathbf{B})^{T} = \begin{pmatrix} 6 & -1 \ 14 & 7 \end{pmatrix} \)
• Individually transpose \( \mathbf{A} \) and \( \mathbf{B} \) and add them: \( \begin{pmatrix} 6 & -1 \ 14 & 7 \end{pmatrix} = \mathbf{A}^{T} + \mathbf{B}^{T} \)

This property is very useful, particularly in simplifying complex matrix expressions in linear algebra problems.

Linear Algebra

Linear algebra is a branch of mathematics dealing with vectors, vector spaces, and matrices, among other topics. It provides the foundation for various practical applications ranging from computer graphics to machine learning. One of the central concepts in linear algebra is understanding how to manipulate and utilize matrices to solve problems effectively.
A matrix is a rectangular arrangement of numbers, symbols, or expressions in rows and columns. Linear algebra focuses on operations such as addition, subtraction, scalar multiplication, and more. A key operation in linear algebra is the transposition of matrices, where you flip a matrix over its diagonal. This operation is crucial in simplifying computations and solving linear equations.

• Understanding matrix dimensions and compatibility is essential for any operation.
• Matrix operations like addition require matrices of identical dimensions, while others like multiplication can be more flexible.
• Linear algebra's principles help in various fields like network theory, statistics, and even quantum mechanics.

Mastering these concepts in linear algebra will significantly aid in solving mathematical problems efficiently and is fundamental in understanding many modern scientific and engineering problems.