Problem 3 - Advanced Engineering Mathematics

Question

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 matrix size is \( 3 \times 3 \).

Step 1: Identify the Rows

Observe the matrix and count the number of rows. A row in a matrix is a horizontal line of numbers. In this matrix, there are 3 horizontal lines.

Step 2: Identify the Columns

Look at the matrix to count the number of columns. A column is a vertical line of numbers. In this matrix, there are 3 vertical lines.

Step 3: Determine the Matrix Size

Combine the number of rows and columns to state the size of the matrix. The size of a matrix is given as \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns. Here, the matrix size is \( 3 \times 3 \).

Key Concepts

Rows and Columns

In a matrix, understanding the difference between rows and columns is crucial. Each set of horizontally aligned numbers in a matrix makes up a row. Think of rows as layers or levels of information stacked on top of each other. Columns, on the other hand, are vertical alignments of numbers. Visualize columns as pillars or towers standing side by side. When counting rows and columns, it can help to use the grid-like structure of the matrix. For the given matrix:\[\left(\begin{array}{rrr} 1 & 2 & -1 \ 0 & 7 & -2 \ 0 & 0 & 5 \end{array}\right)\]- **Rows**: Here, there are three horizontal lines. So, this matrix has 3 rows.- **Columns**: Similarly, there are three vertical alignments of numbers, indicating the matrix has 3 columns. Recognizing rows and columns helps in understanding other matrix properties and operations efficiently.

Matrix Dimensions

Matrix dimensions refer to the size of a matrix, expressed by the number of rows and columns it contains. It is usually denoted as \( m \times n \), where \( m \) is the number of rows, and \( n \) is the number of columns.Let's consider the example matrix:\[\left(\begin{array}{rrr} 1 & 2 & -1 \ 0 & 7 & -2 \ 0 & 0 & 5 \end{array}\right)\]This matrix has 3 rows and 3 columns, making its dimensions \( 3 \times 3 \).Knowing the dimensions of a matrix is key in operations such as addition, subtraction, and multiplication with other matrices. It confirms compatibility between matrices for such operations. For example:- Matrices can be added or subtracted directly if they share the same dimensions.- Matrix multiplication is possible only when the number of columns in the first matrix matches the number of rows in the second.

Matrix Notation

Matrix notation is a systematic way of representing matrices, which helps in carrying out mathematical operations precisely. In matrix notation, a matrix is commonly represented using capital letters, like \( A \), \( B \), or \( C \). Each entry, or element, in the matrix can be specified with a subscript notation. Consider a matrix \( A \):\[A = \left(\begin{array}{rrr} 1 & 2 & -1 \ 0 & 7 & -2 \ 0 & 0 & 5 \end{array}\right)\]**Subscript Notation**:- The element of a matrix at the first row and second column is often addressed as \( a_{1,2} = 2 \). This notation helps to specify the exact position of elements within a matrix without ambiguity. Matrix notation is also essential in explaining transformations in linear algebra, solving systems of equations, and more. By following a consistent notation format, one can easily communicate complex matrix operations and results clearly.