Problem 48 - Discrete Mathematics with Applications

Question

Find each product. $$\left[\begin{array}{rrr} 1 & 1 & 1 \\ 1 & -2 & 3 \\ 2 & -3 & 4 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]$$

Step-by-Step Solution

Answer

The product of the given matrices is: $$ \textbf{AB}=\left[\begin{array}{l} x + y + z \\\ x - 2y + 3z \\\ 2x - 3y + 4z \end{array}\right] $$

Step 1: Set up the matrices

We are given the matrices: $$ \textbf{A} =\left[\begin{array}{rrr} 1 & 1 & 1 \\\ 1 & -2 & 3 \\\ 2 & -3 & 4 \end{array}\right], \quad \textbf{B} =\left[\begin{array}{l} x \\\ y \\\ z \end{array}\right] $$ We need to find the product \(\textbf{AB}\).

Step 2: Multiply the matrices

To multiply the matrices, we will take the dot product of each row of matrix \(\textbf{A}\) with the column of matrix \(\textbf{B}\). The first entry in the resulting matrix is given by: \(1\cdot x + 1\cdot y + 1 \cdot z = x + y + z\) The second entry in the resulting matrix is given by: \(1\cdot x - 2 \cdot y + 3 \cdot z = x - 2y + 3z\) The third entry in the resulting matrix is given by: \(2\cdot x - 3\cdot y + 4\cdot z = 2x - 3y + 4z\)

Step 3: Write down the result

After multiplying the matrices, we get the resulting 3x1 matrix (column vector): $$ \textbf{AB}=\left[\begin{array}{l} x + y + z \\\ x - 2y + 3z \\\ 2x - 3y + 4z \end{array}\right] $$

Key Concepts

Dot Product

The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. This operation is fundamental in the procedure of matrix multiplication. Let's say we have two vectors, \( \textbf{u} = [u_1, u_2, u_3] \) and \( \textbf{v} = [v_1, v_2, v_3] \) then their dot product is calculated by the formula \(\textbf{u} \cdot \textbf{v} = u_1 \times v_1 + u_2 \times v_2 + u_3 \times v_3\).

When used in the context of matrix multiplication, the dot product is performed between a row vector from one matrix and a column vector from another. In each calculation, you multiply respective entries and sum them up to result in a single entry in the product matrix. The process is repeated for each pair of row and column vectors, eventually forming a new matrix. This method is key to understanding how to multiply matrices and manipulate data in various fields of science and engineering, including Discrete Mathematics.

Column Vector

In the realm of matrices, a column vector is a matrix with a single column and multiple rows. They are essential in representing vectors in linear algebra and are extensively used in different calculations. A typical column vector can be represented as \( \textbf{v} = \begin{bmatrix} v_1 \ v_2 \ \vdots \ v_n \end{bmatrix} \) where \(\textbf{v} \) consists of \{n\} elements arranged vertically.

When matrix multiplication is performed, as shown in the given exercise, we multiply a 3x3 matrix with a 3x1 column vector. The product of this operation is another column vector with dimensions 3x1. Each entry in the resultant column vector is calculated by taking the dot product of each row of the matrix with the single column vector, which involves working with each component of the column vector sequentially.

Discrete Mathematics

Discrete Mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. Unlike real numbers that have a property of varying 'smoothly', the objects studied in Discrete Mathematics – such as integers, graphs, and statements in logic – do not vary smoothly, but have distinct, separated values. Discrete Mathematics includes a wide variety of topics such as combinatorics, graph theory, and algorithm analysis, and it is the backbone of computer science and information theory.

Matrix operations, such as matrix multiplication, play a critical role in Discrete Mathematics, particularly in topics related to linear algebra, coding theory, and graph theory. When students encounter matrix multiplication in their studies, they're learning a computational technique that applies across many discrete structures. Understanding the process, which involves concepts like the dot product and the manipulation of column vectors, is essential for deeper studies in the field of Discrete Mathematics.