Problem 2 - College Algebra with Modeling and Visualization

Question

State the dimension of each matrix. \(\begin{array}{ll}\text { (a) }\left[\begin{array}{llll}-1 & 1\end{array}\right] & \text { (b) }\left[\begin{array}{rr}1 & -1 \\ 7 & 5 \\\ -4 & 0\end{array}\right]\end{array}\) (c) \(\left[\begin{array}{rrrr}1 & 3 & 8 & -3 \\ 1 & -1 & 1 & -2 \\ 4 & 5 & 0 & -1\end{array}\right]\)

Step-by-Step Solution

Answer

(a) 1x2; (b) 3x2; (c) 3x4

Step 1: Understand Matrix Dimensions

The dimensions of a matrix are given as \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns.

Step 2: Determine Dimensions of Matrix (a)

Matrix (a) is \([-1, 1]\), which has 1 row and 2 columns. Therefore, the dimension of Matrix (a) is \(1 \times 2\).

Step 3: Determine Dimensions of Matrix (b)

Matrix (b) is \[\begin{bmatrix}1 & -1 \7 & 5 \-4 & 0\end{bmatrix}\] It has 3 rows and 2 columns. Thus, its dimension is \(3 \times 2\).

Step 4: Determine Dimensions of Matrix (c)

Matrix (c) is \[\begin{bmatrix}1 & 3 & 8 & -3 \1 & -1 & 1 & -2 \4 & 5 & 0 & -1\end{bmatrix}\] It has 3 rows and 4 columns. So, the dimension of Matrix (c) is \(3 \times 4\).

Key Concepts

Matrices

A matrix is a rectangular array of numbers, symbols, or expressions organized in rows and columns. These elements are placed in square brackets for easy identification.
Matrix elements are essential in both mathematics and data analysis, serving numerous purposes in calculations.
Different kinds of matrices, such as row matrices and column matrices, have specific structures useful in various applications.

• Row Matrix: A matrix with only one row.
• Column Matrix: A matrix with only one column.
• Square Matrix: A matrix with the same number of rows and columns.

Each of these types supports different mathematical operations, making matrices versatile tools in fields such as statistics, engineering, and computer graphics.

Rows and Columns

Rows and columns are the key components of a matrix. Understanding how these two elements work is crucial for determining the matrix's dimension.
Rows are horizontal collections of elements, while columns are vertical. Counting the number of rows and columns directly reveals the dimensions.
For example, if you have a matrix of dimensions \(m \times n\), you count \(m\) rows and \(n\) columns.

• Matrix (a) represented as \([-1, 1]\) has 1 row and 2 columns, hence a dimension of \(1 \times 2\).
• Matrix (b), being \(\begin{bmatrix}1 & -1 \ 7 & 5 \ -4 & 0\end{bmatrix}\), contains 3 rows and 2 columns, equaling a dimension of \(3 \times 2\).
• Matrix (c), shown as \(\begin{bmatrix}1 & 3 & 8 & -3 \ 1 & -1 & 1 & -2 \ 4 & 5 & 0 & -1\end{bmatrix}\), holds 3 rows and 4 columns, leading to a \(3 \times 4\) dimension.

These dimensions are fundamental to understanding the matrix's structure and how it can be manipulated or used in equations. This understanding will also help in identifying appropriate operations to perform on the matrix.

Matrix Analysis

Matrix analysis involves the study and manipulation of matrices to solve mathematical problems.
It includes operations like addition, subtraction, multiplication, and even more complex tasks like finding determinants or inverses of square matrices.
In order to perform these operations correctly, it's important to first understand the matrix's dimensions and structure.

• Addition and Subtraction: Only possible if matrices have the same dimensions.
• Multiplication: Requires the number of columns in the first matrix to equal the number of rows in the second.
• Determinants/Inverses: Applicable only to square matrices, where rows equal columns.

These operations allow scientists and engineers to handle complex scenarios, such as data transformations or solving linear equations, more effectively.