Problem 9 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine the dimension of the matrix. $$\left[\begin{array}{r} 4 \\ 32 \\ 3 \end{array}\right]$$

Step-by-Step Solution

Answer

The dimension of the given matrix is '3 x 1'. This means it has 3 rows and 1 column.

Step 1: Identify the matrix

Given the matrix: \( \left[\begin{array}{r} 4 \ 32 \ 3 \end{array}\right] \)

Step 2: Determine the number of rows

Count the number of horizontal lines or rows. The given matrix has 3 rows.

Step 3: Determine the number of columns

Count the number of vertical lines or columns. The given matrix has 1 column.

Step 4: Express the dimension of matrix

The dimension of a matrix is represented as 'rows x columns'. Therefore, the dimension of the given matrix is '3 x 1'.

Key Concepts

Understanding Matrices

A matrix is a fundamental concept in algebra that represents a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Think of it like a spreadsheet where each entry of data has its own 'cell'. The order of a matrix is given by the number of rows and columns it contains.

Matrices are used for various applications in mathematics, science, engineering, and computer science, such as solving systems of linear equations, transforming geometric shapes, and even in the field of economics for modeling and analyzing financial data.

In our example, the given matrix is a vertical array of numbers, which can be called a column matrix. To fully understand, visualize this matrix as a narrow skyscraper where each floor represents a number in the matrix. Each 'floor' is one row, and since there's only one 'skyscraper', there's just one column.

Rows and Columns

The structure of a matrix is defined by its rows and columns. In essence, rows are horizontal line-ups of elements within the matrix while columns are vertical. For a quick tip on remembering which is which, think of the word 'row' rhyming with 'low'—indicating a horizontal direction, and the first letter in 'column' could stand for 'climb', suggesting a vertical direction.

To identify the size of a matrix, it's as simple as counting these horizontal and vertical lines. Start from the top left element and count each line right for rows, and each line down for columns. Our example is a straightforward case: it's a tall and narrow matrix with numbers lined up vertically. This arrangement means there are multiple rows, but only a single column.

Basics of Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It allows us to create equations and expressions that can be solved to find unknown values.

Understanding the dimension of matrices is a part of algebra where you're dealing with quantities arranged in rows and columns. When we express the dimension of a matrix, we're essentially explaining its shape in terms of algebraic structure. In our case, with a '3 x 1' matrix, algebra tells us that we can perform certain operations with this matrix, like adding it to another '3 x 1' matrix or multiplying it by a '1 x n' matrix, where 'n' stands for the number of columns of the second matrix.

Algebra with matrices can become complex, but the foundational idea is simple. It starts with recognizing patterns and using consistent rules to describe and solve mathematical problems.