Problem 4 - College Algebra

Question

In Exercises \(1-4\) a. Give the order of each matrix. b. If \(A=\left[a_{i j}\right],\) identify \(a_{32}\) and \(a_{23}\), or explain why identification is not possible. $$\left[\begin{array}{rrrr}-4 & 1 & 3 & -5 \\\2 & -1 & \pi & 0 \\\1 & 0 & -e & 5\end{array}\right]$$

Step-by-Step Solution

Answer

The order of the matrix is 3 x 4. The elements \(a_{32}\) and \(a_{23}\) are 0 and \(\pi\) respectively.

Step 1: Determine the Order of the Matrix

The given matrix has three rows and four columns, so the order of the matrix is 3 x 4.

Step 2: Identify \(a_{32}\) and \(a_{23}\)

\(a_{32}\) refers to the element in the 3rd row and 2nd column. Looking at the matrix, \(a_{32}=0\). Then look for \(a_{23}\) which refers to the element in the 2nd row and 3rd column. Checking the matrix, \(a_{23}=\pi\).

Step 3: Confirm Findings

The order of the matrix is 3 x 4. The element \(a_{32}\) is 0 and the element \(a_{23}\) is \(\pi\).

Key Concepts

Matrix Elements Identification

Understanding how to identify elements in a matrix is crucial for any student delving into linear algebra or related mathematics. Each value in a matrix is referred to as an element. These elements are typically denoted by a letter with two subscripts; in our case, we use 'A' with 'ij' as subscripts, where 'i' and 'j' correspond to the position of the row and column, respectively.

For example, if we take the matrix given in the exercise, we have an array of numbers where each is distinctively placed in a grid format. To identify a specific element like \(a_{32}\), we look for the element that's positioned in the third row and the second column. Why third and second, you ask? It's because the first subscript '3' denotes the row, while the second subscript '2' denotes the column. Applying this rule, \(a_{32}\) in our matrix exercise is 0. Similarly, \(a_{23}\) would be the element in the second row, third column, which is identified as \(\text{\(\pi\)}\) in our example.

Once you grasp this system of labelling, it practically unlocks your ability to navigate any matrix, no matter its size!

Row and Column Notation

Row and column notation is a simple yet efficient way to navigate through the matrix's structure. Whenever we list the dimensions of a matrix, we always go in the order of 'rows x columns'. In the step by step solution for the given exercise, identifying the matrix's order was the first step. The matrix is described as having '3 x 4' order, which means it has three rows and four columns.

The term 'order' here is synonymous with the matrix's size or dimensions. It's also worth mentioning that a matrix can be square (same number of rows and columns) or rectangular (different number of rows and columns), as in the provided example.

Familiarizing with the Grid

Imagine a matrix as a grid where each cell can be located using two coordinates: one for the row and one for the column, following the format ('row number', 'column number'). To practice, one could sketch a simple matrix and label the rows and columns, then try identifying elements using their 'row-column' positions.

Mathematical Constants in Matrices

Encountering mathematical constants in matrices, like \(\pi\) and \(e\), is common in higher-level mathematics and physics problems. In the context of matrices, these constants are treated just like any other element in terms of their positioning.

In the given exercise, the number \(\pi\) is a mathematical constant representing the ratio of a circle's circumference to its diameter and is approximately equal to 3.14159. The constant \(e\) is another indispensable number known as Euler's number, approximately equal to 2.71828, which often appears in natural logarithms and exponential growth problems.

Constants as Elements

In our matrix, \(\pi\) and \(e\) are found as \(a_{23}\) and \(a_{33}\), respectively. This illustrates that constants are placed in matrices akin to any numerical value and can be identified using the same 'row-column' notation. For instance, control systems and quantum mechanics are two fields where you might regularly encounter such matrices inclusive of mathematical constants.