Problem 11 - Algebra 2

Question

State the dimensions of each matrix. $$ \left[\begin{array}{lll}{0} & {0} & {8} \\ {6} & {2} & {4} \\ {1} & {3} & {6} \\ {5} & {9} & {2}\end{array}\right] $$

Step-by-Step Solution

Answer

4 rows and 3 columns, dimensions are 4x3.

Step 1: Identify Rows

Examine the matrix given: \[\begin{bmatrix}0 & 0 & 8 \6 & 2 & 4 \1 & 3 & 6 \5 & 9 & 2\end{bmatrix}\]Count the number of horizontal lines of numbers from top to bottom. There are 4 rows.

Step 2: Identify Columns

Next, count the number of vertical lines of numbers from left to right. There are 3 columns.

Step 3: State Dimensions

The dimensions of a matrix are given in the format \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns. For this matrix, there are 4 rows and 3 columns, so the dimensions are \( 4 \times 3 \).

Key Concepts

Matrix Rows

Matrix rows are the horizontal arrays of numbers that run from left to right across a matrix. Think of rows as lines of seats in a cinema, stretching from one side of the room to the other. In mathematical terms, each row in a matrix is comprised of elements from different columns but located on the same horizontal line.
To identify the number of rows in a matrix, simply count the horizontal lines. For the given example matrix:

• 0, 0, 8
• 6, 2, 4
• 1, 3, 6
• 5, 9, 2

There are 4 distinct lines or rows. Each row contains a set of numbers, making it straightforward to determine how many rows a matrix contains by looking at its structure from top to bottom.

Matrix Columns

Matrix columns are vertical sets of numbers that stretch from the top of the matrix to the bottom. Picture columns as tall towers in a city skyline, each standing next to the other. In a mathematical matrix, each column groups numbers from the various rows but along the same vertical line.
To find out how many columns a matrix has, count the lines of numbers vertically. For our matrix example, consider these groupings:

• Column 1: 0, 6, 1, 5
• Column 2: 0, 2, 3, 9
• Column 3: 8, 4, 6, 2

There are 3 columns in total. This count reveals not just the number of columns, but also hints at how data is organized in a matrix from left to right.

Matrix Notation

Matrix notation is the format used to describe the size or dimensions of a matrix. When working with matrices, it's essential to accurately communicate their size using a standardized format. This notation is expressed in the form \( m \times n \), where \( m \) represents the number of rows, and \( n \) indicates the number of columns.
For example, the matrix from the exercise has 4 rows and 3 columns, so its dimensions are expressed as \( 4 \times 3 \). This concise format provides all the information necessary to understand the structure and size of the matrix.
The notation serves as a quick reference, ensuring everyone is on the same page regarding the shape of the matrix, which is crucial when performing operations like addition, multiplication, or inversion.