Problem 21 - Advanced Engineering Mathematics

Question

$$ \begin{aligned} &\text { If } \mathbf{A}=\left(\begin{array}{r} 4 \\ 8 \\ -10 \end{array}\right) \text { and } \mathbf{B}=\left(\begin{array}{lll} 2 & 4 & 5 \end{array}\right), \text { find }(\mathbf{a}) \mathbf{A}^{T} \mathbf{A},(\mathbf{b}) \mathbf{B}^{T} \mathbf{B} \text { , }\\\ &\text { (c) } \mathbf{A}+\mathbf{B}^{T} \end{aligned} $$

Step-by-Step Solution

Answer

\( \mathbf{A}^T \mathbf{A} = 180, \mathbf{B}^T \mathbf{B} = \begin{pmatrix} 4 & 8 & 10 \\ 8 & 16 & 20 \\ 10 & 20 & 25 \end{pmatrix}, \mathbf{A} + \mathbf{B}^T = \begin{pmatrix} 6 \\ 12 \\ -5 \end{pmatrix} \).

Step 1: Calculate \( \mathbf{A}^T \mathbf{A} \)

To find \( \mathbf{A}^T \mathbf{A} \), first transpose \( \mathbf{A} \) to get \( \mathbf{A}^T = \begin{pmatrix} 4 & 8 & -10 \end{pmatrix} \). Then, multiply \( \mathbf{A}^T \) with \( \mathbf{A} \): \[ \mathbf{A}^T \mathbf{A} = \begin{pmatrix} 4 & 8 & -10 \end{pmatrix} \begin{pmatrix} 4 \ 8 \ -10 \end{pmatrix} \].Perform the multiplication: \[ 4 \times 4 + 8 \times 8 + (-10) \times (-10) = 16 + 64 + 100 = 180 \].Thus, \( \mathbf{A}^T \mathbf{A} = 180 \).

Step 2: Calculate \( \mathbf{B}^T \mathbf{B} \)

Transpose \( \mathbf{B} \) to get \( \mathbf{B}^T = \begin{pmatrix} 2 \ 4 \ 5 \end{pmatrix} \) and multiply by \( \mathbf{B} \): \[ \mathbf{B}^T \mathbf{B} = \begin{pmatrix} 2 \ 4 \ 5 \end{pmatrix} \begin{pmatrix} 2 & 4 & 5 \end{pmatrix} \].Perform the matrix multiplication: \[ \begin{pmatrix} 2 \times 2 & 2 \times 4 & 2 \times 5 \ 4 \times 2 & 4 \times 4 & 4 \times 5 \ 5 \times 2 & 5 \times 4 & 5 \times 5 \end{pmatrix} = \begin{pmatrix} 4 & 8 & 10 \ 8 & 16 & 20 \ 10 & 20 & 25 \end{pmatrix} \].Thus, \( \mathbf{B}^T \mathbf{B} = \begin{pmatrix} 4 & 8 & 10 \ 8 & 16 & 20 \ 10 & 20 & 25 \end{pmatrix} \).

Step 3: Calculate \( \mathbf{A} + \mathbf{B}^T \)

Make sure \( \mathbf{A} \) and \( \mathbf{B}^T \) can be added by verifying they are the same dimensions. Both \( \mathbf{A} = \begin{pmatrix} 4 \ 8 \ -10 \end{pmatrix} \) and \( \mathbf{B}^T = \begin{pmatrix} 2 \ 4 \ 5 \end{pmatrix} \) are column vectors with 3 entries, so they can be added.Add corresponding elements:\[ \mathbf{A} + \mathbf{B}^T = \begin{pmatrix} 4 + 2 \ 8 + 4 \ -10 + 5 \end{pmatrix} = \begin{pmatrix} 6 \ 12 \ -5 \end{pmatrix} \].Therefore, \( \mathbf{A} + \mathbf{B}^T = \begin{pmatrix} 6 \ 12 \ -5 \end{pmatrix} \).

Key Concepts

Matrix Transposition

Matrix transposition is a simple, yet vital operation in linear algebra. It involves flipping a matrix over its diagonal, effectively swapping its row and column indices. Consider matrix \( \mathbf{A} \), represented as \( \begin{pmatrix} 4 \ 8 \ -10 \end{pmatrix} \). The transpose of \( \mathbf{A} \), denoted as \( \mathbf{A}^T \), converts the column matrix into a row matrix, resulting in \( \mathbf{A}^T = \begin{pmatrix} 4 & 8 & -10 \end{pmatrix} \).

• Each element \( a_{ij} \) in the original becomes \( a_{ji} \) in the transposed.
• In reality, this operation is necessary for operations like matrix multiplication, where the alignment of rows and columns matters.

Transposing vectors makes them conform to matrix multiplication rules, ensuring their dimensions are compatible. Practicing this operation increases familiarity, making advanced problems more approachable.

Matrix Multiplication

Matrix multiplication is an operation where two matrices, or a matrix and a vector, combine to form another matrix. In our exercise, we dealt with \( \mathbf{A}^T \mathbf{A} \) and \( \mathbf{B}^T \mathbf{B} \).

For \( \mathbf{A}^T \mathbf{A} \), after transposing \( \mathbf{A} \):

• The resulting multiplication is \( \mathbf{A}^T \mathbf{A} = \begin{pmatrix} 4 & 8 & -10 \end{pmatrix} \begin{pmatrix} 4 \ 8 \ -10 \end{pmatrix} \).
• This involves summing the products of corresponding elements: \( 4 \times 4 + 8 \times 8 + (-10) \times (-10) = 180 \).

This calculation yielded a scalar. Likewise, for \( \mathbf{B}^T \mathbf{B} \):

• The transpose and multiplication resulted in a 3x3 matrix, each entry obtained by multiplying rows by columns and summing.

Matrix multiplication isn’t just about numbers; it models transformations, crucial in fields like physics and engineering.

3D Vectors

3D vectors are a cornerstone of mathematics and physics, representing points or directions in three-dimensional space. In the exercise, vectors \( \mathbf{A} \) and \( \mathbf{B} \) were involved. Each vector accounts for entities in a 3D coordinate system. For example, \( \mathbf{A} = \begin{pmatrix} 4 \ 8 \ -10 \end{pmatrix} \) represents a point or position in 3D space with x, y, and z coordinates.

When performing operations such as addition, like in \( \mathbf{A} + \mathbf{B}^T \), piecewise addition occurs:

• The x-components add together, y-components with y, z-components with z.
• Result: \( \begin{pmatrix} 4 + 2 \ 8 + 4 \ -10 + 5 \end{pmatrix} = \begin{pmatrix} 6 \ 12 \ -5 \end{pmatrix} \).

3D vectors are essential for simulating real-world phenomena and are used extensively in computer graphics, navigation, and more. Understanding their manipulation allows one to engage deeper with spatial problems.