Question

Determine the order of the matrix. $$ \left[\begin{array}{rr} 33 & 45 \\ -9 & 20 \\ 12 & 15 \\ 16 & -2 \end{array}\right] $$

Short answer

The order of the given matrix is 4 x 2.

Step-by-step solution

Count the Rows

Count the rows in the matrix. A row is a horizontal line, and in this matrix, there are 4 rows: \[33, 45]\, [-9, 20]\, [12, 15]\, [16, -2]. Thus this matrix has 4 rows.

Count the Columns

Count the columns in the matrix. A column is a vertical line. In this matrix, there are 2 columns: the first one is \[33, -9, 12, 16] and the second one is \[45, 20, 15, -2]. Hence, this matrix has 2 columns.

Find the Order

The order of a matrix is defined as 'rows x columns'. Here, there are 4 rows and 2 columns. Therefore, the order of this matrix is '4 x 2'.

Key concepts

Matrices in Algebra

When you study algebra, you'll soon encounter a powerful mathematical tool called a matrix. Matrices are arrays of numbers or expressions arranged in rows and columns, which can represent data, equations, transformations in geometry, and more. They are enclosed in square or rectangular brackets and are essential in various areas of mathematics and science.

Algorithms that include matrix operations are at the heart of computer graphics, physics simulations, and statistical analysis. The idea is to organize information in a way that allows for efficient calculation and transformation. With a clear understanding of matrix basics, you'll unlock the full potential of this mathematical representation in solving complex problems and equations in algebra and beyond.

Rows and Columns in Matrices

Understanding the layout of matrices is crucial, and this is where 'rows' and 'columns' come into play. Each row in a matrix is a set of numbers that run horizontally, while each column contains numbers running vertically. Think of a matrix as a spreadsheet, where each row represents a record and each column represents a different field of information.

Operations like addition, subtraction, and multiplication heavily depend on the rows and columns of the matrices involved. For instance, when multiplying two matrices, the number of columns in the first matrix must match the number of rows in the second. By recognizing this structure, you'll be better equipped to manipulate matrices effectively.

Matrix Dimensions

The 'order' or 'dimensions' of a matrix refers to the number of rows and columns it contains, always listed in that sequence. It's written in the format 'rows' by 'columns', like 'm x n' for a matrix with 'm' rows and 'n' columns. This is fundamental when identifying the size of a matrix and determining the compatibility of matrices for certain operations.

The order tells you how many data points you're working with, and whether or not you can apply specific procedures like matrix multiplication. Getting the dimensions right will save you time and eliminate errors in your computations. So, every time you work with matrices, start by identifying their order—it's a step you simply can't skip.