Problem 1 - Advanced Engineering Mathematics

Question

In Problems 1-20, fill in the blanks or answer true/false. A matrix \(\mathbf{A}=\left(a_{i j}\right)_{4 \times 3}\) such that \(a_{i j}=i+j\) is given by _________.

Step-by-Step Solution

Answer

The matrix is \[\mathbf{A} = \begin{bmatrix} 2 & 3 & 4 \\ 3 & 4 & 5 \\ 4 & 5 & 6 \\ 5 & 6 & 7 \end{bmatrix}\].

Step 1: Understanding the Matrix Dimensions

The matrix \(\mathbf{A}\) is given as a \(4 \times 3\) matrix. This means it has 4 rows and 3 columns. Our objective is to determine each element of this matrix based on the rule provided.

Step 2: Analyzing the Element Formula

Each element \(a_{ij}\) of the matrix is defined by the formula \(i + j\). Here, \(i\) represents the row number and \(j\) represents the column number. Therefore, we will calculate each element with this formula.

Step 3: Calculating Each Element in the Matrix

For each element in the \(4 \times 3\) matrix, calculate \(a_{ij} = i + j\):- Row 1: - \(a_{11} = 1+1 = 2\) - \(a_{12} = 1+2 = 3\) - \(a_{13} = 1+3 = 4\)- Row 2: - \(a_{21} = 2+1 = 3\) - \(a_{22} = 2+2 = 4\) - \(a_{23} = 2+3 = 5\)- Row 3: - \(a_{31} = 3+1 = 4\) - \(a_{32} = 3+2 = 5\) - \(a_{33} = 3+3 = 6\)- Row 4: - \(a_{41} = 4+1 = 5\) - \(a_{42} = 4+2 = 6\) - \(a_{43} = 4+3 = 7\).

Step 4: Writing the Complete Matrix

Now, write down the entire matrix using the calculated elements:\[\mathbf{A} = \begin{bmatrix} 2 & 3 & 4 \3 & 4 & 5 \4 & 5 & 6 \5 & 6 & 7 \end{bmatrix}\]

Key Concepts

Understanding Matrix Dimensions

In matrix algebra, understanding the dimensions of a matrix is crucial. Matrix dimensions are expressed as rows by columns. This notation gives the matrix its shape. For instance, the given matrix \(\mathbf{A}\) is said to be a \(4 \times 3\) matrix. This means it has 4 rows and 3 columns.

• The first number always represents the number of rows.
• The second number represents the number of columns.

Knowing these dimensions helps us better organize and reference elements within the matrix. For students learning matrix formations, recognizing dimensions is the first stepping stone in manipulating and understanding matrices.

Making Sense of Matrix Elements

Matrix elements are the individual numbers within a matrix, each identified by their position. In our example with matrix \(\mathbf{A}\), each element is represented as \(a_{ij}\), where \(i\) is the row number, and \(j\) is the column number.

These positions determine the specific value of each element, according to a given rule or formula. In this exercise, the formula used is \(i + j\). This means:

• For an element in the first row, second column (\(a_{12}\)), \(i=1\) and \(j=2\), making the element value \(1 + 2 = 3\).
• For an element in the third row, first column (\(a_{31}\)), \(i=3\) and \(j=1\), the element value is \(3 + 1 = 4\).

Understanding how these elements are calculated is essential for interpreting and creating matrices based on specific conditions or rules.

The Process of Matrix Formation

Matrix formation is constructing a matrix by filling in its elements based on a given rule or formula. In this task, the matrix \(\mathbf{A}\) is formed by using the rule \(a_{ij} = i + j\). This means each element is calculated by adding its row and column indices.

Here’s how you can form a \(4 \times 3\) matrix:

• Start by counting the rows and columns.
• Use the formula to determine the value for each element.
• Fill in each position within the matrix as calculated.

The result is a matrix where every element follows the given pattern, ensuring consistency and understanding of how matrices operate. Proper matrix formation is foundational in higher-level math, ensuring clarity in applications like computer graphics, statistics, and many more fields.