Problem 1 - Advanced Engineering Mathematics

Question

In Problems \(1-6\), state the size of the given matrix. $$ \left(\begin{array}{llll} 1 & 2 & 3 & 9 \\ 5 & 6 & 0 & 1 \end{array}\right) $$

Step-by-Step Solution

Answer

The matrix size is 2x4.

Step 1: Identify the Dimensions of the Matrix

First, identify the number of rows and columns in the given matrix. The matrix is written as two sets of horizontal entries organized in rows, and these rows consist of vertical arrangements called columns.

Step 2: Determine the Number of Rows

Count how many horizontal arrays (rows) exist. In this matrix, there are 2 rows: \(\begin{array}{llll}1 & 2 & 3 & 9 \end{array}\) and \(\begin{array}{llll}5 & 6 & 0 & 1\end{array}\).

Step 3: Determine the Number of Columns

Count how many columns are present by examining any single row. Each row has four numbers, indicating that there are 4 columns: 1, 2, 3, and 9 form the columns.

Step 4: Formulate the Matrix Size

Combine the row and column counts to express the size of the matrix. The matrix has 2 rows and 4 columns, making it a 2x4 matrix.

Key Concepts

Matrix Rows

Matrix rows are essentially the horizontal sections of a matrix. In any given matrix, the rows are the horizontal lines of numbers. Each number in a row represents an element of the row. Think of each row as a list where items are lined up side by side.

To determine the number of rows in a matrix, you simply count how many lines of elements there are. For example, in the matrix you are analyzing: \[\left(\begin{array}{llll} 1 & 2 & 3 & 9 \ 5 & 6 & 0 & 1 \end{array}\right)\]this matrix has 2 rows.

Understanding the rows is fundamental because the first number in a matrix size notation (like "2x4") represents the number of rows. So here, the '2' refers to the two rows present.

Matrix Columns

Columns in a matrix are the vertical stacks that run top to bottom. Like rows, each column in a matrix contains a lineup of numbers or elements.

To find out how many columns there are in a matrix, you can choose any single row and count the number of elements it has. Since all rows must have the same number of columns, this count gives you the total number of columns in the matrix.

In the matrix: \[\left(\begin{array}{llll} 1 & 2 & 3 & 9 \ 5 & 6 & 0 & 1 \end{array}\right)\]there are 4 columns because each row has 4 elements. Columns are crucial in defining a matrix's structure, and when we say a matrix is 2x4, the '4' denotes the number of columns.

Matrix Size Calculation

Matrix size calculation is the method used to define the dimensions of a matrix. It's expressed in the form of 'rows x columns'.

To calculate the size of a matrix, begin by counting the number of rows. Once you know the row count, count the number of columns using any row.

For the matrix provided: \[\left(\begin{array}{llll} 1 & 2 & 3 & 9 \ 5 & 6 & 0 & 1 \end{array}\right)\]there are 2 rows and 4 columns. Therefore, the size of this matrix is 2x4. Remembering the order—rows first, then columns—helps maintain accuracy in matrix identification and manipulation.

Understanding matrix dimensions is key in many mathematical contexts, as it affects operations like matrix addition, multiplication, and graph plotting.