Problem 33 - Advanced Engineering Mathematics

Question

$$, 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. $$ (\mathbf{A B})^{T}=\mathbf{B}^{T} \mathbf{A}^{T} $$

Step-by-Step Solution

Answer

The property \((\mathbf{AB})^{T} = \mathbf{B}^{T} \mathbf{A}^{T}\) is verified, as both sides equal \( \begin{pmatrix} 16 & -10 \\ 40 & -20 \end{pmatrix} \).

Step 1: Compute Matrix AB

To find \( \mathbf{A} \times \mathbf{B} \), we perform the matrix multiplication as follows:\( \mathbf{A} = \begin{pmatrix} 2 & 4 \ -3 & 2 \end{pmatrix} \)\( \mathbf{B} = \begin{pmatrix} 4 & 10 \ 2 & 5 \end{pmatrix} \)\[ \mathbf{A} \times \mathbf{B} = \begin{pmatrix} (2)(4) + (4)(2) & (2)(10) + (4)(5) \ (-3)(4) + (2)(2) & (-3)(10) + (2)(5) \end{pmatrix} = \begin{pmatrix} 16 & 40 \ -10 & -20 \end{pmatrix} \]

Step 2: Transpose of Matrix AB

To find the transpose \( \mathbf{(AB)}^{T} \), interchange the rows and columns of the product matrix:\[ \begin{pmatrix} 16 & 40 \ -10 & -20 \end{pmatrix}^{T} = \begin{pmatrix} 16 & -10 \ 40 & -20 \end{pmatrix} \]

Step 3: Transpose of B and A

Compute the transpose of matrices \( \mathbf{B} \) and \( \mathbf{A} \):\[ \mathbf{B}^{T} = \begin{pmatrix} 4 & 2 \ 10 & 5 \end{pmatrix} \]\[ \mathbf{A}^{T} = \begin{pmatrix} 2 & -3 \ 4 & 2 \end{pmatrix} \]

Step 4: Multiply Transposed Matrices (B^T A^T)

Now, multiply the transposed matrices \( \mathbf{B}^{T} \) and \( \mathbf{A}^{T} \):\( \mathbf{B}^{T} = \begin{pmatrix} 4 & 2 \ 10 & 5 \end{pmatrix} \)\( \mathbf{A}^{T} = \begin{pmatrix} 2 & -3 \ 4 & 2 \end{pmatrix} \)\[ \mathbf{B}^{T} \times \mathbf{A}^{T} = \begin{pmatrix} (4)(2) + (2)(4) & (4)(-3) + (2)(2) \ (10)(2) + (5)(4) & (10)(-3) + (5)(2) \end{pmatrix} = \begin{pmatrix} 16 & -10 \ 40 & -20 \end{pmatrix} \]

Step 5: Verify the Equality

Compare the results of the two calculated matrices \( \mathbf{(AB)}^{T} \) and \( \mathbf{B}^{T} \mathbf{A}^{T} \):Both matrices are \( \begin{pmatrix} 16 & -10 \ 40 & -20 \end{pmatrix} \).This confirms the property \( (\mathbf{AB})^{T} = \mathbf{B}^{T} \mathbf{A}^{T} \) holds.

Key Concepts

Matrix Multiplication

Matrix multiplication is an essential operation in linear algebra where two matrices are multiplied together. This process does not involve multiplying individual elements by their counterparts directly. Instead, it involves a specific pattern.
To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. In the exercise provided, matrices \( \mathbf{A} \) and \( \mathbf{B} \) are compatible for multiplication because \( \mathbf{A} \) is a 2x2 matrix and \( \mathbf{B} \) is a 2x2 matrix too.
The resulting product matrix's element at position \((i, j)\) is the sum of the products of corresponding elements from the \(i\)-th row of the first matrix and the \(j\)-th column of the second matrix.

• For example, element (1,1) of the product matrix in our problem is calculated as: \( (2 \times 4) + (4 \times 2) = 16 \).
• Similarly, element (2,2) is: \( (-3 \times 10) + (2 \times 5) = -20 \).

Mastering matrix multiplication is crucial for various applications in engineering, physics, computer science, and more.

Transpose of a Matrix

Transposing a matrix means flipping it over its diagonal, essentially swapping rows with columns. This operation converts the \(i\)-th row of a matrix into the \(i\)-th column of its transpose.
For instance, if you have a matrix \( \mathbf{C} \) with dimensions m x n, then the transpose, \( \mathbf{C}^{T} \), will be n x m. In simple terms, the first row of \( \mathbf{C} \) becomes the first column of \( \mathbf{C}^{T} \).

• In our case, matrix \((\mathbf{AB})\) is transposed to form \((\mathbf{AB})^{T}\), changing from \(\begin{pmatrix} 16 & 40 \ -10 & -20 \end{pmatrix} \) to \(\begin{pmatrix} 16 & -10 \ 40 & -20 \end{pmatrix} \).

Transposing matrices is a common operation in many mathematical computations and helps relate different matrix properties, as highlighted in this exercise.

Matrix Properties

Matrices have various intriguing properties that can simplify complex mathematical expressions and proofs. One fundamental property is the transpose of a product. In mathematical terms, if you multiply two matrices \( \mathbf{A} \) and \( \mathbf{B} \), the transpose of their product \( (\mathbf{AB})^{T} \) is equal to the product of their transposes in reverse order, \( \mathbf{B}^{T} \mathbf{A}^{T} \).
In the given exercise, this property was demonstrated by calculating both \( (\mathbf{AB})^{T} \) and \( \mathbf{B}^{T}\mathbf{A}^{T} \) separately and confirming their equality. This property is especially useful when manipulating large matrices or solving systems of equations.
Understanding these properties helps shed light on matrix behavior in theoretical math and practical applications like computer graphics and data transformations.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, matrices, and vector spaces. It's a foundational discipline for any math or engineering student, as it provides essential tools for scientific computations and theoretical developments.
Matrix operations such as multiplication and transposition are core aspects of linear algebra, commonly used in algorithm design, physics simulations, and more.
Linear algebra helps simplify complex problems by modeling them in higher dimensions, allowing for easier computations and deeper understanding of multidimensional data. This makes matrix operations like those performed in the exercise not only relevant for academic exercises, but also essential for real-world problem-solving in fields like artificial intelligence, economics, and network theory.