Problem 2 - Advanced Engineering Mathematics

Question

In Problems \(1-6\), state the size of the given matrix. $$ \left(\begin{array}{ll} 0 & 2 \\ 8 & 4 \\ 5 & 6 \end{array}\right) $$

Step-by-Step Solution

Answer

The matrix is 3x2.

Step 1: Identify the Matrix

The given matrix is a collection of numbers organized in rows and columns inside parentheses or brackets. Here, the matrix is presented as: \[\left(\begin{array}{ll} 0 & 2 \ 8 & 4 \ 5 & 6 \end{array}\right)\] which signifies the arrangement of numbers in rows and columns.

Step 2: Count the Rows

The first step in determining the size of a matrix is to count the number of rows. Observe the horizontal alignment of numbers. In this matrix, the rows are:- Row 1: \([0, 2]\)- Row 2: \([8, 4]\)- Row 3: \([5, 6]\)There are 3 rows in total.

Step 3: Count the Columns

Next, count the number of columns. Columns are arranged vertically. In this matrix, each row has 2 elements, indicating two columns:- Column 1: \([0, 8, 5]\)- Column 2: \([2, 4, 6]\)There are 2 columns in total.

Step 4: State the Size of the Matrix

The size of the matrix is given as 'm x n', where 'm' is the number of rows and 'n' is the number of columns. From the steps above, the matrix has 3 rows and 2 columns. Therefore, the size of the matrix is 3x2.

Key Concepts

Matrix Rows

When exploring the concepts of a matrix, the first thing to understand is what matrix rows are. Rows in a matrix are the horizontal lines of numbers or variables arranged from left to right. Think of rows as the horizontal structure upon which a matrix is built. Each row contains a collection of elements within the matrix.
For example, in the given matrix: \[ \begin{array}{ll} 0 & 2 \ 8 & 4 \ 5 & 6 \end{array} \]we can identify three distinct rows:

• Row 1: \( [0, 2] \)
• Row 2: \( [8, 4] \)
• Row 3: \( [5, 6] \)

By counting these rows, we can determine the number of rows in the matrix, which is essential for specifying the matrix size.

Matrix Columns

Columns in a matrix are equally important as rows. They refer to the vertical alignment of the matrix elements. Each column comprises elements taken from each row and stacked in a vertical manner.
In the same matrix:\[ \begin{array}{ll} 0 & 2 \ 8 & 4 \ 5 & 6 \end{array} \]we find two columns by looking at the vertical arrangement:

• Column 1: \( [0, 8, 5] \)
• Column 2: \( [2, 4, 6] \)

Recognizing and counting these columns helps determine another part of the matrix size. In this case, there are two columns.

Matrix Dimensions

Understanding the dimensions of a matrix is crucial when working with matrices. The term 'matrix dimensions' refers to the number of rows and columns that a matrix contains. It's expressed as 'm x n' where 'm' represents the number of rows and 'n' represents the number of columns.
For example, if we consider the given matrix:\[ \begin{array}{ll} 0 & 2 \ 8 & 4 \ 5 & 6 \end{array} \]We found that it has:

• 3 rows
• 2 columns

Hence, its dimensions can be stated as 3x2. Recognizing these dimensions is fundamental in matrix operations like addition, subtraction, and multiplication, which require matching specific dimension rules.