Problem 14 - Precalculus with Limits

Question

In Exercises 9-14, determine the order of the matrix. \( \left[\begin{array}{rrr} -7 & & 6 & & 4\\\ 0 & & -5 & & 1 \end{array}\right] \)

Step-by-Step Solution

Answer

The order of the given matrix is 2 x 3.

Step 1: Determine the Rows

Begin by counting the number of rows in the matrix. A row in a matrix is a horizontal line of elements. The matrix has two rows: \[ \begin{array}{ccc} -7 & 6 & 4 \ 0 & -5 & 1 \end{array} \]

Step 2: Determine the Columns

Next, count the number of columns in the matrix. A column in a matrix is a vertical line of elements. The given matrix has three columns: \[ \begin{array}{ccc} -7 & 6 & 4 \ 0 & -5 & 1 \end{array} \]

Step 3: Formulate the Order

Having determined that there are two rows and three columns, the order of the matrix can thus be described as '2 x 3'.

Key Concepts

What is a Matrix?

Understanding matrices is essential in various fields such as mathematics, engineering, and computer science. A matrix is a rectangular array of numbers or other mathematical objects, arranged in rows and columns. Each individual number within a matrix is known as an element. For example, if we have the matrix \[\begin{array}{ccc}-7 & 6 & 4 \0 & -5 & 1\end{array}\],
the elements are \(-7\text{, }6\text{, }4\text{, }0\text{, }-5\text{, and }1\).
Matrices are used to solve systems of linear equations, represent transformations, and carry out tasks in applied mathematics. The utility of matrices is wide-ranging and spans across theoretical and practical applications alike.

Rows and Columns in Matrices

Each matrix is defined by its rows and columns. Rows run horizontally and are often considered as individual vectors within the matrix. In the example given, the matrix has two rows:

• The first row is \(-7\text{, }6\text{, }4\)
• The second row is \(0\text{, }-5\text{, }1\)

Columns, on the other hand, run vertically. They also can be seen as individual vectors that represent the matrix in another dimension. In the presented example, the matrix has three columns:

• The first column is \(-7\text{, }0\)
• The second column is \(6\text{, }-5\)
• The third column is \(4\text{, }1\)

Understanding the structure of rows and columns is vital for operations such as matrix addition, subtraction, and multiplication by another matrix or a scalar.

Matrix Dimensions

The dimensions of a matrix, also known as its 'order', are given by the number of rows and columns it contains. The format used to describe the dimensions is 'rows x columns'. For example, a matrix with 2 rows and 3 columns is said to have an order of 2x3. The matrix in our problem:
\[\begin{array}{ccc}-7 & 6 & 4 \0 & -5 & 1\end{array}\]
has two rows and three columns, so we can say its order is 2x3. This notation helps in identifying the type and size of the matrix at a glance, which is crucial when performing matrix operations where the dimensions must match specific criteria for the operations to be viable.