Problem 8 - Algebra 2

Question

Refer to matrices \(A\) and \(B\) at the right. Identify each matrix element. \(A=\left[\begin{array}{rr}{0} & {-1} \\ {1.5} & {3} \\ {7} & {-2}\end{array}\right] \quad B=\left[\begin{array}{lll}{6} & {-3} & {\frac{1}{2}}\end{array}\right]\) \(a_{31}\)

Step-by-Step Solution

Answer

The element \(a_{31}\) of Matrix A is 7.

Step 1: Understand Matrix Element Notation

Matrix elements are denoted by a lowercase letter corresponding to the matrix, followed by two subscript numbers. The first subscript indicates the row number, and the second subscript indicates the column number of the element within the matrix.

Step 2: Locate the Element in Matrix A

Since we are looking for the element labelled as \(a_{31}\), this refers to the element in the third row and the first column of Matrix A.

Step 3: Identify the Value

By locating the third row and the first column of Matrix A, we find that the value of the element \(a_{31}\) is 7.

Key Concepts

Matrix Notation

When dealing with matrices, it's crucial to understand the conventional method used to reference its elements, commonly referred to as matrix notation. A matrix is essentially a rectangular array of numbers arranged in rows and columns. These numbers are called elements.

To identify each element within the matrix, we use a notation that combines a lowercase letter representing the matrix, usually the letter at the beginning of the alphabet like 'a' for Matrix A, with a pair of subscript numbers. For example, in the expression \( a_{ij} \), 'a' signifies the matrix's name, and 'i' and 'j' represent the row and column of the specific element within the matrix, respectively.

Understanding this notation is the first step in being able to navigate and manipulate matrices, which are fundamental components in various fields of science and engineering, including computer science, physics, and economics.

Subscript Numbers

Subscript numbers play a defining role in matrix notation. They precisely indicate the location of an element within a matrix. The first subscript number, which we typically denote as 'i', corresponds to the row of the matrix. Rows are horizontal arrays of numbers and are counted from top to bottom starting with the number one.

The second subscript number, represented by 'j', pinpoints the column of the matrix. Columns are vertical stacks within the matrix, and like rows, they are also counted from left to right beginning with one. Therefore, the notation \( a_{23} \) would indicate the element found at the intersection of the second row and third column of Matrix A. Such specific markers are crucial when working with larger matrices or conducting operations like matrix multiplication where precise element alignment is needed.

Algebraic Matrices

Algebraic matrices are at the heart of linear algebra and are used to solve systems of linear equations, transform geometric data, and model real-world problems in various sciences. As seen in the earlier example with matrices A and B, an algebraic matrix is a set of numbers or functions laid out in a rectangular grid format.

These values are not just random; they can represent data points, coefficients of equations, or other significant numerical information in a compact and structured way. Operations on algebraic matrices, such as addition, subtraction, and multiplication, follow specific rules that take into account the positions of the elements, which are identified using the subscript notation. It's vital for students to not only learn how to perform these operations but also understand their implications and how they correspond to various algebraic problems and applications.