Problem 12 - Algebra 2

Question

State the dimensions of each matrix. $$ \left[\begin{array}{rrr}{-3} & {17} & {-22} \\ {9} & {31} & {16} \\ {20} & {-15} & {4}\end{array}\right] $$

Step-by-Step Solution

Answer

The dimensions of the matrix are 3x3.

Step 1: Identify Rows and Columns

To determine the dimensions of a matrix, identify how many rows and how many columns it has. Remember that rows are horizontal lines of numbers, and columns are vertical.

Step 2: Count the Rows

Count the number of horizontal lines in the matrix. In this case, the matrix has 3 rows.

Step 3: Count the Columns

Count the number of vertical lines in the matrix. In this case, the matrix has 3 columns.

Step 4: State the Dimensions

Use the format 'rows x columns' to express the dimensions of the matrix. Since this matrix has 3 rows and 3 columns, the dimensions are 3x3.

Key Concepts

Understanding Matrices

A matrix is essentially a rectangular array of numbers arranged in rows and columns. You encounter matrices in a multitude of math and science subjects because they are a compact way to represent complex data. Matrices are particularly significant in the fields of algebra, physics, and computer science, facilitating operations like transformations, rotations, and solving linear equations. Matrices can vary in size. It's their dimensions, given by the number of rows and columns, that define their size and shape. Each element within the matrix is typically denoted as a number, and the position of the element is determined by its row and column indices. Understanding their dimensions helps you perform mathematical operations such as addition, subtraction, and multiplication.

Defining Rows in a Matrix

Rows in a matrix are the horizontal sequences of elements. When you look at a matrix, the numbers aligned from left to right constitute a row. Counting the rows will give you the first part of a matrix's dimension. In the given example, the matrix displays the following rows:

• -3, 17, -22
• 9, 31, 16
• 20, -15, 4

It's essential to visualize each row as a separate line of data. Rows can often represent individual records, different time states, or unique cases, depending on the context in which the matrix is used. Distinguishing between rows and columns is crucial for correct matrix manipulation and computation.

Distinguishing Columns in Matrices

Columns in a matrix are the vertical sequences of numbers. When examining a matrix, each number that follows downwards forms a column. Similar to rows, counting the columns results in the second part of the matrix's dimension notation. For clarity, let's represent the columns from the given matrix:

• Column 1: -3, 9, 20
• Column 2: 17, 31, -15
• Column 3: -22, 16, 4

Columns can represent different variables, various attributes or multiple conditions. Identifying columns correctly is vital when performing operations like matrix multiplication, where the interaction between columns of one matrix and rows of another is fundamental.

Matrices in Algebra

In algebra, matrices are foundational elements for numerous computations. They serve as building blocks for solving systems of equations, performing coordinate transformations, and much more. Matrices can also store coefficients of linear equations or data for statistical analysis. Algebraic operations involving matrices include addition, subtraction, and multiplication, all of which require a fundamental understanding of matrix dimensions. For instance, two matrices can only be added or subtracted if they share identical dimensions, meaning they must have the same number of rows and columns. Matrix multiplication, on the other hand, involves multiplying rows of one matrix with columns of another, and this can only occur when the number of columns in the first matrix matches the number of rows in the second. Matrices, by offering a structured and visual approach to hold information, allow for various transformations and computations that would be cumbersome with pure arithmetic. As such, they simplify and solve complex algebraic problems efficiently.