Verified Answer :

  • Identify the rows: The matrix has 3 horizontal lines of elements, so there are 3 rows.
  • Identify the columns: The matrix has 2 vertical lines of elements, so there are 2 columns.
  • State the dimensions: The dimensions of the matrix are expressed as \(\text{rows} \times \text{columns}\).
  • Final Answer: \(\boxed{3 \times 2}\)

Explanation

  1. Understanding Matrix Dimensions

The matrix provided is:
\[ \begin{bmatrix} 6 & -7 \ -5 & -1 \ 0 & 4 \end{bmatrix} \]
To determine the dimensions of a matrix, we need to count the number of horizontal rows and the number of vertical columns. The dimensions are always expressed as the number of rows followed by the number of columns, denoted as \((\text{rows}) \times (\text{columns})\).

  1. Counting Rows and Columns

First, we identify the rows. By observing the horizontal arrangement of the elements, we can see:

  • Row 1: \(6, -7\)
  • Row 2: \(-5, -1\)
  • Row 3: \(0, 4\)

There are 3 rows in total.

Next, we identify the columns. By observing the vertical arrangement of the elements, we can see:

  • Column 1: \(6, -5, 0\)
  • Column 2: \(-7, -1, 4\)

There are 2 columns in total.

  1. Stating the Dimensions

Since the matrix has 3 rows and 2 columns, its dimensions are \(3 \times 2\). The final result is \(\boxed{3 \times 2}\).

Examples

Matrix dimensions are essential in many fields, such as computer graphics, data analysis, and engineering. For example, if you are organizing test scores for a class where each student is a row and each subject is a column, a \(3 \times 2\) matrix would represent 3 students' scores across 2 subjects. Understanding these dimensions allows for efficient data storage and helps in performing matrix operations like addition or multiplication.

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