Problem 6 - Precalculus: Mathematics for Calculus

Question

State the dimension of the matrix. $$\left[\begin{array}{rrrr} -1 & 5 & 4 & 0 \\ 0 & 2 & 11 & 3 \end{array}\right]$$

Step-by-Step Solution

Answer

The dimension of the matrix is \(2 \times 4\).

Step 1: Identify the Rows

First, let's count the number of rows in the given matrix. A row is a horizontal line of elements. In the given matrix, we have two horizontal lines: \([-1, 5, 4, 0]\) and \([0, 2, 11, 3]\). Thus, there are 2 rows.

Step 2: Identify the Columns

Next, count the number of columns in the matrix. A column is a vertical line of elements. Consider each position across the row lines: \([-1,0]\), \([5,2]\), \([4,11]\), and \([0,3]\). There are 4 such columns.

Step 3: State the Matrix Dimension

The dimension of a matrix is given in the format \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns. From the previous steps, we have 2 rows and 4 columns, so the matrix dimension is \(2 \times 4\).

Key Concepts

Rows and Columns in a Matrix

Understanding rows and columns is fundamental to understanding matrices. A matrix is a rectangular array of numbers arranged in horizontal lines known as rows and vertical lines referred to as columns. In the matrix \(\left[\begin{array}{rrrr} -1 & 5 & 4 & 0 \ 0 & 2 & 11 & 3 \end{array}\right]\), you can identify rows and columns quickly:

• A row collects elements horizontally. In this example, the matrix has two rows: the first row \([-1, 5, 4, 0]\) and the second row \([0, 2, 11, 3]\).
• A column assembles elements vertically. Here, the columns are \([-1, 0]\), \([5, 2]\), \([4, 11]\), and \([0, 3]\).

Identifying the structure of rows and columns will allow you to describe the matrix's dimensions and perform other matrix operations effectively.

Matrix Identification

Matrix identification is the process of determining the size or dimension of a matrix, typically described in the format \(m \times n\), where \(m\) stands for the number of rows and \(n\) signifies the number of columns. In our example, we examined the matrix \( \left[\begin{array}{rrrr} -1 & 5 & 4 & 0 \ 0 & 2 & 11 & 3 \end{array}\right] \) and established that there are 2 rows and 4 columns. Therefore, this matrix is a \(2 \times 4\) matrix.


Correctly identifying the matrix size serves multiple purposes.

• It helps in conforming matrices for addition or multiplication, as these operations require certain dimension compatibility.
• It is essential for the application of many mathematical algorithms that involve matrices.

Learning this skill will allow you to tackle more complex matrix-related problems with ease.

Types of Matrices

Matrices can vary in their design and purpose, classified into different types based on dimensions and element values. Here’s a few commonly encountered types:

• Square Matrix: A matrix with the same number of rows and columns, like a \(3 \times 3\) matrix. These are vital in defining matrix determinants and eigenvalues.
• Row Matrix: Consists of a single row. For example, a \(1 \times n\) matrix is termed a row matrix.
• Column Matrix: Consists of a single column. Like a row matrix, a column matrix is of form \(m \times 1\).
• Zero Matrix (or Null Matrix): A matrix in which every element is zero, which can be any dimension, such as \(2 \times 2\) or \(3 \times 4\).
• Diagonal Matrix: A square matrix in which non-diagonal elements are zero. All the key values lie on the diagonal from the upper left to the lower right.

Recognizing the type of matrix helps clarify what operations or transformations can be effectively employed. For instance, only square matrices have inverses or determinants, which is critical in solving linear equations.