Problem 53 - Algebra 2

Question

State the dimensions of each matrix. $$ \left[\begin{array}{rrr}{7} & {-3} & {5} \\ {0} & {2} & {-9} \\ {6} & {5} & {1}\end{array}\right] $$

Step-by-Step Solution

Answer

The matrix is 3x3.

Step 1: Identify the Rows

Count the number of horizontal lines of numbers in the matrix. There are 3 rows in this matrix.

Step 2: Identify the Columns

Count the number of vertical lines of numbers in the matrix. There are 3 columns in this matrix.

Step 3: State the Dimensions

The dimensions of a matrix are given in the form of the number of rows by the number of columns. Here, the matrix has 3 rows and 3 columns, so the dimensions are 3x3.

Key Concepts

Rows in a Matrix

In the world of matrices, understanding rows is crucial. Rows are the horizontal arrangements of numbers within a matrix. Think of them like shelves in a bookcase, where each shelf is a separate row. For example, in a matrix such as the one given: \[\begin{bmatrix} 7 & -3 & 5 \ 0 & 2 & -9 \ 6 & 5 & 1 \end{bmatrix}\] there are three rows. Each row consists of numbers lined up horizontally. The numbers 7, -3, and 5 make up the first row. Similarly, the second row consists of 0, 2, and -9, and the third row has 6, 5, and 1. When identifying rows:

• Count each horizontal line once, from left to right.
• These rows help determine one part of the matrix's dimensions.

Understanding rows lays a foundation for comprehending how matrices function.

Columns in a Matrix

Columns in a matrix are just as important as rows. They are the vertical lines of numbers. Imagine them as stacks of canned goods at a grocery store. Each stack represents a column in the matrix. Referring back to the given matrix: \[\begin{bmatrix} 7 & -3 & 5 \ 0 & 2 & -9 \ 6 & 5 & 1 \end{bmatrix}\] this matrix has three columns. The first column contains the numbers 7, 0, and 6. The second column is made up of -3, 2, and 5, and the third column consists of the numbers 5, -9, and 1. When counting columns:

• Look at each vertical line, moving from top to bottom.
• The number of these columns makes up the second dimension in the notation.

By understanding columns, you can describe the matrix more precisely.

Matrix Notation

Matrices, fundamentally, are represented in a specific format called matrix notation. This notation expresses the entire size or dimension of the matrix through a simple format, such as "3x3", derived from the number of rows and columns. For example, consider the matrix: \[\begin{bmatrix} 7 & -3 & 5 \ 0 & 2 & -9 \ 6 & 5 & 1 \end{bmatrix}\] This matrix is described using the notation '3x3', meaning it has three rows and three columns. This is critical because:

• It provides a quick, concise view of the matrix's shape.
• Understanding this notation ensures clear communication regarding matrix operations and solutions.

Matrix notation simplifies discussions about matrices and helps you quickly comprehend their structure.