Problem 6 - Advanced Engineering Mathematics

Question

State the size of the given matrix. $$ \left(\begin{array}{r} 1 \\ 5 \\ -6 \\ 0 \\ 7 \\ -10 \\ 2 \\ 12 \end{array}\right) $$

Step-by-Step Solution

Answer

8 × 1

Step 1: Understand Matrix Layout

A matrix is organized in rows and columns. The size of the matrix is given by the number of rows and the number of columns, which are usually indicated as rows × columns.

Step 2: Identify Number of Rows

Examine the given matrix, which is written in a vertical arrangement. Count how many numbers are stacked one over the other to determine the number of rows. For the given matrix: 1 5 -6 0 7 -10 2 12 Count of numbers (or lines): 8. Hence, the matrix has 8 rows.

Step 3: Identify Number of Columns

The given matrix only has one column, as all numbers are aligned in a single vertical line.

Step 4: Conclude the Matrix Size

With the rows determined as 8 and columns determined as 1, the size of the matrix is denoted as 8 × 1.

Key Concepts

Matrix Notation

Matrix notation is a specific way of writing and organizing data in mathematics. Think of a matrix as somewhat like a table. It has rows and columns to display information systematically. In mathematical terms, a matrix is denoted by a set of square brackets or parentheses, within which the data is listed. Each element of the matrix is enclosed inside these brackets. If you're visualizing matrices, consider them as grid-like structures, even if, sometimes, they might just appear as rows or just as columns. For instance, our example matrix is written vertically, which makes it look like a list, but using the brackets indicates that it is indeed a matrix.

Matrix Row Count

The row count of a matrix signifies how many horizontal entries the matrix contains. This implies counting how many separate lines of numbers appear in a matrix. To determine the row count, simply scan the lines from top to bottom. In our expressed vertical matrix:

• Each number appears on a new line.
• We have a total of 8 numbers.

Thus, this matrix contains 8 rows. Recognizing the number of rows is vital, as it forms part of the matrix's dimensions, revealing part of how it is structured.

Matrix Column Count

Columns in a matrix represent vertical entries or stacks of elements aligned one above another. Whenever you see a number of values lined in vertical order without deviation to left or right, they are potentially in one column. For the provided matrix:

• Every value is written in one continuous vertical line.
• This indicates that we have only 1 column.

Understanding the column count is essential to fully grasp the structure of the matrix, as it is a principal component in defining a matrix's dimensions.

Matrix Dimensions

Matrix dimensions give us the complete size description by combining rows and column counts. They help us convey the physical shape of the matrix. This is often expressed as "rows × columns." For instance, the matrix provided in the exercise has:

• 8 rows
• 1 column

Therefore, its dimensions are written as 8 × 1. This highlights that the matrix spans vertically with eight levels of data in a single straight line. Knowing a matrix's dimensions is crucial, as it provides a clear specification of its capacity to hold and structure data within a mathematical or computational task.