Problem 30 - Algebra 2

Question

Which element in matrix \(A=\left[\begin{array}{llll}{5} & {-2} & {-\frac{1}{2}} & {0}\end{array}\right]\) is the number 0\(?\) A. \(a_{40}\) B. \(a_{4}\) C. \(a_{14}\) D. \(a_{41}\)

Step-by-Step Solution

Answer

\(a_{14}\)

Step 1: Understand Matrix Indexing

Matrix elements are indexed using two numbers. The first number represents the row, and the second number represents the column.

Step 2: Determine the Position of the Element

Since the matrix provided is a 1x4 matrix, we look at the first (and only) row for the number 0. Its position is the 4th column.

Step 3: Identify the Correct Matrix Index

The element 0 is at index 14, which in matrix notation is written as the row number followed by the column number. In this case, it is denoted as \(a_{14}\).

Key Concepts

Matrices in Algebra

In algebra, matrices are powerful tools that store numbers and variables in an organized, rectangular array. These arrays allow us to perform various operations such as addition, subtraction, and multiplication of matrices, as well as more complex actions like finding determinants and inverses. Matrices are applied in solving systems of linear equations, transforming geometric figures, and representing data in various fields including computer graphics, statistics, and engineering.

Understanding matrices in algebra involves comprehending their structure and the different types, including square, rectangular, row, and column matrices. The ability to manipulate matrices is cornerstone in linear algebra, and it plays a critical role in further mathematical studies and practical applications.

Matrix Notation

Matrix notation is a formal way of describing the arrangement of elements within a matrix. Each element in a matrix is denoted by a letter, often 'a', followed by two subscripts. The first subscript denotes the row, and the second denotes the column. For example, in matrix \(A\), the element \(a_{23}\) refers to the element in the second row and third column.

Proper notation becomes more critical as matrices increase in size and complexity. It helps in uniquely identifying each element, which is essential for matrix operations like addition, where elements from corresponding positions are added together, or multiplication, where dot products of rows and columns are used. Notation simplifies the communication of the structure and specific elements within a matrix, ensuring clarity in mathematical expressions and problem-solving.

Math Problem Solving

Math problem solving is a step-by-step approach to understanding and solving mathematical questions. It starts with understanding the problem, then devising a plan on how to tackle it. Following the plan, you solve for the solution, and finally, review if the result makes sense in the context of the original problem.

For matrix-related problems, key skills involve identifying the structure of the matrix, understanding matrix notation, and applying appropriate algebraic rules to find the solution. Problem-solving in linear algebra can range from simple tasks, like locating a particular element, to more complex operations such as matrix inversion or eigenvalue calculation. It's crucial to be methodical and logical in each step to ensure a successful outcome. Students should practice these skills with various types of matrix problems to develop their mathematical reasoning and precision.