Problem 52 - Algebra 2

Question

State the dimensions of each matrix. $$ \left[\begin{array}{rrrr}{5} & {1} & {-6} & {2} \\ {-38} & {5} & {7} & {3}\end{array}\right] $$

Step-by-Step Solution

Answer

The matrix dimensions are 2 by 4.

Step 1: Identify rows

Count the number of horizontal arrays (rows) in the matrix. This matrix has 2 horizontal arrays: the first one is \([5, 1, -6, 2]\) and the second is \([-38, 5, 7, 3]\). Therefore, there are 2 rows.

Step 2: Identify columns

Count the number of vertical arrays (columns) in the matrix. Each row has four elements, which together form four columns. Therefore, there are 4 columns.

Step 3: State the dimensions

Combine the number of rows and columns to state the dimensions of the matrix. The dimensions are stated as rows by columns, so the dimensions of the matrix are 2 by 4.

Key Concepts

matrix

A matrix is essentially a rectangular array of numbers, symbols, or expressions set out by rows and columns. It has a structured format which helps to represent data neatly and efficiently. The elements within a matrix are usually enclosed with square brackets. For example, the matrix from the problem contains elements showcased in an ordered way with two horizontal lists (rows) and four vertical lists (columns). Matrices are widely used in mathematics and related fields to solve equations, transform data, and more. Key features of a matrix:

• Arrays of elements aligned in a grid-like pattern.
• Arranged into rows and columns, making it easy to identify dimensions.
• Elements can be numbers, variables, or even expressions.

rows and columns

When working with matrices, it's imperative to understand the concepts of rows and columns.

Rows

Rows in a matrix are horizontal arrays of elements. Each list of numbers extending from left to right on the matrix's horizontal line counts as a row. In the provided exercise's matrix, the rows are \([5, 1, -6, 2] \)and \([-38, 5, 7, 3]\).

Columns

Columns are vertical arrays of elements in a matrix. Columns run from top to bottom, and each row in the matrix contributes one element to a column. In the example, because there are four numbers in each row, it confirms the existence of four distinct columns.The concept of rows and columns are fundamental in determining the matrix's size or dimension, with dimensions written as ‘rows by columns’.

matrices

Matrices are fundamental elements in various scientific and engineering disciplines due to their versatility and function. They assist in organizing complex data into an understandable structure. Important aspects of matrices:

• Dimensions: This refers to the number of rows and columns within a matrix, defined as 'rows by columns'. In practical applications, matrices offer insight into data dimensions, much like coordinate systems.
• Practical Applications: They serve as tools in solving systems of equations, perform transformations in computer graphics, and help in data organization in fields like statistics and computer science.

Matrices can vary greatly in size and shape, being as small as a single element or extending to vastly large arrays to handle complex data synthesis and transformations. Understanding their construction and properties is essential for effective use in mathematics and beyond.