Problem 4 - Advanced Engineering Mathematics

Question

State the size of the given matrix. $$ \left(\begin{array}{lll} 5 & 7 & -15 \end{array}\right) $$

Step-by-Step Solution

Answer

The size of the matrix is 1 x 3.

Step 1: Identify the Matrix Rows and Columns

The given matrix is presented as \( \left( \begin{array}{lll} 5 & 7 & -15 \end{array} \right) \). This is a single horizontal arrangement of elements, referred to as a row. Count the number of distinct rows and columns visible in the matrix.

Step 2: Count the Number of Rows

The matrix contains just one horizontal set of numbers, making it one row. Therefore, the number of rows is 1.

Step 3: Count the Number of Columns

Each distinct element in the horizontal arrangement represents a separate column. Count the numbers: 5, 7, and -15. There are three distinct elements, hence the matrix contains 3 columns.

Step 4: Determine the Size of the Matrix

The size of a matrix is denoted as (rows × columns). The given matrix has been determined to have 1 row and 3 columns. Therefore, the size of the matrix is 1 x 3.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is essential when dealing with matrices. Dimensions tell us the overall structure and layout of a matrix. The dimensions are given by the number of rows and columns present. This is represented in two numbers separated by an "x" symbol, denoted as "rows × columns." For instance, if a matrix has 2 rows and 3 columns, its dimensions are noted as 2 × 3. This notation helps differentiate between various types of matrices, such as row matrices, column matrices, and square matrices.
To determine the dimensions of any given matrix, a straightforward method is followed:

• First, count the number of horizontal rows.
• Then, count the number of vertical columns.
• Combine these counts in the format of rows × columns.

This procedure allows a quick assessment of the matrix size, which is fundamental for performing operations such as matrix multiplication.

Matrix Rows and Columns

Matrices are composed of rows and columns, each playing a significant role in matrix operations. Each row is a horizontal arrangement of elements, while each column is a vertical arrangement. Understanding this structure is crucial for solving problems involving matrices.
Here are key points to help identify and understand rows and columns:

• Rows: Run from left to right horizontally. The number of rows can be counted by observing the horizontal sequences in the matrix.
• Columns: Run from top to bottom vertically. The number of columns can be identified by counting the vertical sequences in the matrix.
• In the given exercise, the matrix \[\left( \begin{array}{lll} 5 & 7 & -15 \end{array} \right)\] is a single row with three columns, making it a row matrix.

Recognizing rows and columns helps in accurately determining a matrix's dimensions and conducting matrix operations.

Matrix Notation

Matrix notation is a fundamental aspect of working with matrices. It provides a clear and standardized way to represent a matrix, which aids in communication and understanding of mathematics. Two major types of notation involve the layout and naming of a matrix.

• Layout: Matrices are arranged in brackets. For example, \[\left( \begin{array}{lll} 5 & 7 & -15 \end{array} \right)\] uses a pair of square brackets with elements placed inside, showing its structure.
• Naming: Matrices are often labeled with capital letters, such as A, B, C, etc. This helps easily reference them in equations and descriptions.
• The size of a matrix, as described earlier, comes into play in notation through representations such as "1 × 3," indicating one row and three columns.

Being familiar with matrix notation is essential for tackling more advanced mathematical problems and ensures clear communication when sharing or discussing matrices.