Problem 6 - Algebra and Trigonometry

Question

State the dimension of the matrix. $$ \left[\begin{array}{rrrr}{-1} & {5} & {4} & {0} \\ {0} & {2} & {11} & {3}\end{array}\right] $$

Step-by-Step Solution

Answer

The dimension of the matrix is 2x4.

Step 1: Understand the Matrix

The given matrix is \( \begin{bmatrix} -1 & 5 & 4 & 0 \ 0 & 2 & 11 & 3 \end{bmatrix} \), which consists of rows and columns.

Step 2: Count the Rows

In the matrix, identify the number of horizontal lines of elements. The matrix has two horizontal lines, indicating that it has 2 rows.

Step 3: Count the Columns

Next, identify the number of vertical arrangements of elements. The matrix has four vertical sets of elements, indicating that it has 4 columns.

Step 4: State the Dimensions

The dimensions of a matrix are given by the number of rows and columns it has. Therefore, the dimensions of this matrix are \(2 \times 4\).

Key Concepts

Matrices

Matrices are collections of numbers arranged in a rectangular format, consisting of rows and columns. They are fundamental in mathematics, especially in areas like linear algebra. Each number within a matrix is known as an element. The purpose of matrices is to provide a structured way to handle and manipulate data and variables for a variety of applications.

• A matrix is defined by its arrangement of rows and columns.
• Each position in the matrix is designated for a single element, often a number.
• Matrices can have different sizes, and this is determined by the number of rows and columns they contain.

Understanding matrices allows you to perform a wide range of mathematical functions, from solving systems of equations to transforming coordinates in geometry. They serve as compact representations of linear transformations and are used extensively in physics, engineering, computer science, and beyond.

Rows and Columns

To understand a matrix's structure, it is essential to differentiate between rows and columns. Rows are the horizontal lines of elements in a matrix, while columns are the vertical lines.

• Rows move from left to right across a matrix, often signifying individual equations or data sets.
• Columns go from top to bottom within the matrix and can represent different variables or dimensions of data.
• The number of rows and columns defines the matrix's size, also known as its dimensions.

For example, the matrix in our exercise \(\left[\begin{array}{rrrr}{-1} & {5} & {4} & {0} \{0} & {2} & {11} & {3}\end{array}\right]\) has 2 rows and 4 columns. Thus, it is referred to as a '2 by 4' matrix. This knowledge is crucial for accurately performing operations such as matrix addition, multiplication, and finding determinants or inverses.

Matrix Notation

Matrix notation is a concise way to describe the structure and elements of a matrix. It uses a standardized form to communicate the size and content of matrices clearly.

• A matrix is typically enclosed within square brackets \(\left[ \cdots \right]\).
• Within the brackets, elements are arranged in rows and columns, separated by space or commas, and each new row is typically written on a new line.
• The order of a matrix is denoted as \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns.

For instance, the matrix depicted in our exercise is written as \(\begin{bmatrix}-1 & 5 & 4 & 0 \0 & 2 & 11 & 3\end{bmatrix}\) with dimensions \(2 \times 4\). Correct matrix notation allows for precise communication and computation, simplifying complex mathematical concepts and aiding in the error-free manipulation of matrices in various mathematical applications.