Question

Determine the order of the matrix. $$ \left[\begin{array}{rrr} 12 & -2 & 4 \\ -3 & 4 & 0 \\ -8 & 12 & 2 \end{array}\right] $$

Short answer

The order of the given matrix is \(3 \times 3\).

Step-by-step solution

Identifying Rows and Columns

Count the number of rows and columns in the matrix. The matrix has 3 rows and 3 columns.

Writing the Order

The order of a matrix is represented as 'm x n', where 'm' stands for the number of rows and 'n' for the number of columns. Here, both 'm' and 'n' are 3. So, the order of the given matrix is '3 x 3'.

Key concepts

Matrix Dimensions

Understanding the dimensions of a matrix is key to many mathematical operations. The dimensions define the size of the matrix and are given as 'm x n', where "m" is the number of rows and "n" is the number of columns. For example, a matrix with 3 rows and 4 columns has dimensions of 3 x 4. This notation helps in understanding how the matrix can be used for operations like addition, multiplication, and determining if it covers specific data.
Remember, the order in which dimensions are stated is important. Rows always come first, followed by columns. Having a clear grasp of matrix dimensions can simplify understanding more complex matrix concepts.

Rows and Columns

To determine the structure of a matrix, you need to identify its rows and columns. A row is a horizontal line of elements, and a column is a vertical line of elements. For instance, in a matrix with 3 rows and 2 columns, there are three separate lines of data horizontally and two vertically.

• Rows: These help in segmenting data in categories that are related horizontally.
• Columns: These are useful for separating data into attributes or variables.

Being able to identify and distinguish between rows and columns enables a deeper understanding of matrix configurations and aids in mathematical problem-solving, such as solving linear equations or performing matrix transformations.

Square Matrix

A special type of matrix is the square matrix. It has the same number of rows and columns, making it symmetrical. For example, a 3 x 3 matrix, like the one provided, is a square matrix. This is important because square matrices have unique properties that are crucial in advanced mathematics.
Square matrices can be used in various scenarios:

• They are essential for finding determinants, which are used in calculating matrix inverses and solving systems of linear equations.
• They facilitate eigenvalue calculations, important in many fields like physics and engineering.

Understanding that a matrix is square opens up more potential operations and solutions, giving students more tools for solving complex problems.