Problem 3 - Advanced Engineering Mathematics

Question

In Problems \(1-6\), state the size of the given matrix. $$ \left(\begin{array}{rrr} 1 & 2 & -1 \\ 0 & 7 & -2 \\ 0 & 0 & 5 \end{array}\right) $$

Step-by-Step Solution

Answer

The size of the matrix is 3x3.

Step 1: Understand the Concept

The size of a matrix is determined by the number of rows and columns it has. The format for describing the size of a matrix is "rows x columns".

Step 2: Count the Rows

Look at the given matrix and count how many rows it has. Rows are horizontal lines of elements.

Step 3: Count the Columns

Next, count how many columns the matrix has. Columns are vertical lines of elements.

Step 4: Determine the Size of the Matrix

Combine the number of rows and columns into the format "rows x columns" to determine the size of the matrix. For example, if there are 3 rows and 3 columns, the matrix size is 3x3.

Key Concepts

Understanding Matrix Rows

Matrix rows are one of the fundamental components that define the structure of a matrix. In a matrix, rows are the horizontal arrays of elements.
Each element within a row is separated by a space or a comma, depending on how the matrix is notated.
When observing a matrix like the one in the exercise, to find the number of rows, you should:

• Look at the lines of numbers going from left to right.
• Count each distinct set of horizontal numbers.

In the given matrix:\[\left(\begin{array}{rrr} 1 & 2 & -1 \ 0 & 7 & -2 \ 0 & 0 & 5 \end{array}\right)\]there are 3 rows:

• \([1, 2, -1]\)
• \([0, 7, -2]\)
• \([0, 0, 5]\)

Understanding rows is crucial because they represent one of two dimensions that determine the matrix's size.

Exploring Matrix Columns

Matrix columns are equally vital as rows in determining the matrix dimensions. Columns are the vertical stacks of elements in a matrix.
Each item in a column aligns vertically with other elements from different rows within the same matrix, helping define its vertical structure.
To count the number of columns in a matrix, such as the provided one, you should:

• Observe the matrix from top to bottom.
• Identify each distinct vertical line of numbers.

In the matrix from the exercise:\[\left(\begin{array}{rrr} 1 & 2 & -1 \ 0 & 7 & -2 \ 0 & 0 & 5 \end{array}\right)\]we find:

• The first column is \([1, 0, 0]\).
• The second column is \([2, 7, 0]\).
• The third column is \([-1, -2, 5]\).

Understanding columns alongside rows provides a complete picture of the matrix's structure, allowing you to accurately describe its dimensions.

Grasping Matrix Dimensions

Matrix dimensions unify the concepts of rows and columns into a cohesive measure of the matrix's size. Dimensions are expressed in the format "number of rows x number of columns". This descriptive format tells you how the matrix lays out in both horizontal and vertical axes.
This is crucial for performing mathematical operations involving matrices as it determines what operations are valid or possible.
In our current example, the matrix\[\left(\begin{array}{rrr} 1 & 2 & -1 \ 0 & 7 & -2 \ 0 & 0 & 5 \end{array}\right)\]exhibits:

• 3 rows, as analyzed in the section on matrix rows.
• 3 columns, as noted in the section on matrix columns.

Therefore, the dimensions are expressed as "3x3" or "three by three". Knowing the matrix dimensions is essential, especially when working with more complex matrix operations or solving systems of linear equations.