Problem 5 - Advanced Engineering Mathematics

Question

State the size of the given matrix. $$ \left(\begin{array}{rrrr} 1 & 5 & -6 & 0 \\ 7 & -10 & 2 & 12 \\ 0 & 9 & 2 & -1 \end{array}\right) $$

Step-by-Step Solution

Answer

The matrix is 3x4.

Step 1: Identify the Number of Rows

Observe the given matrix and count the number of horizontal lines or rows present in the matrix. This matrix has 3 horizontal lines, indicating it has 3 rows.

Step 2: Identify the Number of Columns

Look at the matrix and count the number of vertical lines or columns. This matrix shows 4 vertical groups of numbers, indicating it has 4 columns.

Step 3: Determine the Dimensions of the Matrix

Combine the results from the previous two steps. The size of the matrix is determined by the number of rows followed by the number of columns. Therefore, the matrix has dimensions 3 rows by 4 columns.

Key Concepts

Understanding Rows and Columns in a Matrix

When exploring matrices in mathematics, it's crucial to grasp the concept of rows and columns. A matrix is essentially a rectangular array of numbers. The horizontal lines you see in a matrix are its rows.

• Rows run from left to right across the matrix.
• They form the base structure, making it easy to determine how many data sets we have horizontally.

On the other hand, columns are the vertical lines in a matrix.

• Columns run from top to bottom.
• They stack data points vertically, helping in defining the arrangement of numbers.

Visualizing a spreadsheet can help. Think of rows as the number of different items listed, and columns as each different category or attribute those items share.

Counting the Size of a Matrix

Knowing how to count the size of a matrix is like reading its 'blueprint'. The size is expressed as a pair of numbers.First, count the number of rows, which are the horizontal data sets that span the width of the matrix. Next, count the columns, which are sets of numbers aligned vertically.

The size of a matrix is always noted as 'rows by columns'. In mathematical terms: \[\text{Size of matrix} = m \times n\]where \( m \) is the number of rows, and \( n \) is the number of columns.

• This representation (e.g., 3x4) succinctly provides a snapshot of the matrix's structure.
• This process is straightforward once you know how to count the lines.

Matrix Representation Explained

Matrix representation might initially seem intimidating, but it's merely a way of organizing numbers.

• Each entry in the matrix is contained in a cell formed by the intersection of a row and a column.
• The position of a number within a matrix is key and is usually identified by two indices: row number and column number.
• For example, in a matrix of size \(3 \times 4\), the element in the second row and third column is considered "in position \( (2,3) \)".

These matrices can be used in various applications such as solving simultaneous equations, representing data, and transforming geometrical figures. Whenever you look at a matrix, remember it's organized data waiting to be interpreted or manipulated.